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+Selective Conformal Inference with FCR Control
+Yajie Baoa, Yuyang Huob, Haojie Rena and Changliang Zoub∗
+aSchool of Mathematical Sciences, Shanghai Jiao Tong University
+Shanghai, P.R. China
+bSchool of Statistics and Data Science, Nankai University
+Tianjin, P.R. China
+January 3, 2023
+Abstract
+Conformal inference is a popular tool for constructing prediction intervals (PI). We consider here
+the scenario of post-selection/selective conformal inference, that is PIs are reported only for individuals
+selected from an unlabeled test data. To account for multiplicity, we develop a general split conformal
+framework to construct selective PIs with the false coverage-statement rate (FCR) control. We first
+investigate the Benjamini and Yekutieli (2005)’s FCR-adjusted method in the present setting, and show
+that it is able to achieve FCR control but yields uniformly inflated PIs. We then propose a novel solution
+to the problem, named as Selective COnditional conformal Predictions (SCOP), which entails performing
+selection procedures on both calibration set and test set and construct marginal conformal PIs on the
+selected sets by the aid of conditional empirical distribution obtained by the calibration set. Under
+a unified framework and exchangeable assumptions, we show that the SCOP can exactly control the
+FCR. More importantly, we provide non-asymptotic miscoverage bounds for a general class of selection
+procedures beyond exchangeablity and discuss the conditions under which the SCOP is able to control
+the FCR. As special cases, the SCOP with quantile-based selection or conformal p-values-based multiple
+testing procedures enjoys valid coverage guarantee under mild conditions. Numerical results confirm the
+effectiveness and robustness of SCOP in FCR control and show that it achieves more narrowed PIs over
+existing methods in many settings.
+Keywords: Conditional empirical distribution; Distribution-free; Non-exchangeable conditions; Post-
+selection inference; Prediction intervals; Split conformal.
+∗Corresponding Author: nk.chlzou@gmail.com
+1
+arXiv:2301.00584v1  [stat.ME]  2 Jan 2023
+
+1
+Introduction
+To improve the prediction performance in modern data, many sophisticated machine learning algorithms
+including various “black-box” models are proposed. While often witnessing empirical success, quantifying
+prediction uncertainty is one of the major issues for interpretable machine learning. Conformal inference
+(Vovk et al., 1999, 2005) provides a powerful and flexible tool to quantify the uncertainty of predictions.
+Consider a typical setting that we observe one labeled data set Dl = {(Xi, Yi)}2n
+i=1 and a set of unla-
+belled/test samples Du = {Xi}2n+m
+i=2n+1 whose outcomes {Yi}2n+m
+i=2n+1 are unobserved. Generally, suppose all
+(Xi, Yi) ∈ X × Y are i.i.d from some unknown distribution, and µ(x) := Y | X = x as the prediction model
+associated with (X, Y ), which is usually estimated by the labeled data Dl. For any Xj ∈ Du and a given
+miscoverage level α, standard conformal prediction methods (Lei et al., 2018), yield a prediction interval (PI)
+with distribution-free coverage guarantee, PIα(Xj),
+P(Yj ∈ PIα(Xj)) ≥ 1 − α,
+under independent and identically distributed (i.i.d) (or exchangeable data) assumptions.
+With the development of big data, making predictive inference on all available data (Du) is either
+unnecessary or inefficient in many applications. For example, in the recruitment decisions, only some selected
+viable candidates can get into interview processes (Faliagka et al., 2012; Shehu and Saeed, 2016). In the drug
+discovery trials, researchers select promising ones based on predicting candidates’ activity for further clinical
+trials (Carracedo-Reboredo et al., 2021; Dara et al., 2021). Related applications also appear in financial
+investment and scientific discovery (Jin and Candès, 2022). In such problems, the most common way is to
+select a subset of interest with some rules through some statistical/machine learning algorithms at first, and
+then perform statistical inference only on the selected samples.
+Formally, letting ˆSu ⊆ {2n + 1, . . . , 2n + m} be the selected subset, our goal is to construct the PI of Yj
+for each j ∈ ˆSu. As pointed by Benjamini and Yekutieli (2005), ignoring the multiplicity in construction of
+post-selection intervals will result in distorted average coverage. Under the context of post-selection inference
+in which confidence intervals for multiple selected parameters/variables are being reported, Benjamini and
+Yekutieli (2005) pioneered the criterion, false coverage-statement rate (FCR), to take account for multiplicity.
+The FCR, an analog of the false discovery rate (FDR), can readily be adapted to the present conformal
+inference setting. It is defined as the expected ratio of number of reported PIs failing to cover their respective
+true outcomes to the total number of reported PIs, say
+FCR := E
+�
+|{j ∈ ˆSu : Yj ̸∈ PIj}|
+max{| ˆSu|, 1}
+�
+,
+(1)
+where PIj is the PI for the selected sample j ∈ ˆSu. Benjamini and Yekutieli (2005) provided a selection-
+agnostic method which adjusts the confidence level through multiplying α by a quantity which is related
+to the proportion of selected candidates over all candidates and then constructed the marginal confidence
+2
+
+intervals at the adjusted level for each selected candidate. We will hereafter call it the FCR-adjusted method.
+Accordingly, we may expect that the FCR-adjusted PIs enjoy valid FCR control properties. However, due
+to the dependence structure among PI| ˆ
+Su|α/m(Xj)’s, the results in Benjamini and Yekutieli (2005) are not
+directly applicable in the setting of conformal inference. Please refer to Section 2.1 for detailed discussions
+and rigorous theories. We notice that Weinstein and Ramdas (2020) also discussed the selective inference
+problem under the framework of conformal prediction. The authors suggested to use the FCR-adjusted
+method, however, they did not provide theoretical or empirical investigations.
+While the FCR-adjusted approach can reach FCR control and is widely used, it is generally known to yield
+uniformly inflated confidence intervals (Weinstein et al., 2013). This is because that when calculating the
+noncoverage probabilities of confidence intervals, the adjusted confidence intervals do not take into account the
+selection event. Along this line, Weinstein et al. (2013), Zhao and Cui (2020) and Zhao (2022) further proposed
+some methods to narrow the adjusted confidence intervals by incorporating more selection information. Among
+some others, Fithian et al. (2014), Lee et al. (2016) and Taylor and Tibshirani (2018) proposed constructing
+conditional confidence intervals for each selected variables and showed that the selective error rate can
+be controlled given that the selected set is equal to some deterministic subset. However, those methods
+either require some tractable conditional distribution assumptions or are only applicable for some given
+prediction algorithms, such as normality assumptions or LASSO model, which would limit their applicability
+in the conformal inference. Fortunately, by the virtue of the availability of Dl, distribution/model-agnostic
+conditional prediction intervals with theoretical guarantee can be achieved.
+1.1
+Our contributions
+In this paper, we develop a novel conformal framework to construct post-selection prediction intervals while
+control the FCR, named as Selective COnditional conformal Predictions (SCOP). Our method stems from the
+split conformal inference (Lei et al., 2018; Fithian and Lei, 2020), where the labeled data Dl is split into two
+disjoint parts, one as the training set for obtaining a prediction model ˆµ(X), and the remaining one as the
+calibration set for estimating the distribution of the discrepancy between the Y and ˆµ(X). Then, the key
+ingredient of our proposal entails performing a pre-specified selective procedure on both the calibration set and
+the test set and construct the marginal conformal PIs on the selected sets with the help of conditional empirical
+distribution obtained by the calibration set. The proposed SCOP procedure is model- or distribution-agnostic,
+in the sense that it could wrap around any prediction algorithms with commonly used selection procedures to
+construct PIs.
+The main contributions of the paper can be summarized as follows:
+• Firstly, we investigate the FCR-adjusted method in the setting of conformal inference and show that
+it is able to achieve FCR control under mild conditions, which lays a foundation for our subsequent
+development of SCOP.
+3
+
+• Secondly, under a unified framework and exchangeable assumptions, we show that the SCOP can exactly
+control the FCR at the target level.
+• Thirdly, we provide non-asymptotic miscoverage bounds for a general class of selection procedures beyond
+exchangeablity, termed as ranking-based procedures. This broadens the scopes of our SCOP in theoretical
+guarantee and practical use. To address the non-exchangeability between the the post-selection test set
+and calibration set, we introduce a virtual post-selection calibration set in our proof, and then quantify
+the conditional miscoverage gap between the virtual calibration and the real calibration in SCOP. This
+new technique may be of independent interest for conformal prediction for non-exchangeable data.
+• Finally, we illustrate the easy coupling of the SCOP with commonly used prediction algorithms.
+Numerical experiments indicate that it yields more accurate FCR control than existing methods, while
+offers the narrowed prediction intervals.
+1.2
+Connections to existing works
+Post-selection inference.
+Post-selection inference on a large number of variables has attracted considerable
+research attention. Besides the references mentioned before, a relevant direction is the splitting-based strategy
+for high-dimensional inference. The number of variables is firstly reduced to a manageable size using one
+part of data, while confidence intervals or significance tests can be constructed by computing estimates in a
+low-dimensional region with the other part of data and selected variables. See Wasserman and Roeder (2009),
+Rinaldo et al. (2019), Du et al. (2021) and the references therein. One potential related work is Chen and
+Bien (2020), in which the authors considered to construct confidence intervals for regression coefficients after
+removing the potential outliers from the data. Our paradigm differs substantially with those works as we
+focus on post-selection inference for sample selection rather than variable selection, and existing works on
+variable selection is difficult to extend to the present problem due to the requirements on model or distribution
+assumptions.
+Conformal prediction.
+The building block of our SCOP is the conformal inference framework, which has
+been well studied in many settings, including non-parametric regression (Lei et al., 2013), quantile regression
+(Romano et al., 2019), high-dimensional regression (Lei et al., 2018) and classification (Sadinle et al., 2019;
+Romano et al., 2020), etc. More comprehensive reviews can be found in Shafer and Vovk (2008), Zeni et al.
+(2020) and Angelopoulos and Bates (2021). Conventionally, conformal PIs enjoy distribution-free marginal
+coverage guarantee with the assumption that the data are exchangeable. However, the exchangeability may be
+violated in practice and would be more severe in the post-selection conformal inference because the selection
+procedure might be determined by either the labelled data or the test data, or both. In such situations, one
+particularly difficult issue is that the selected set ˆSu is random and has a complex dependence structure to
+the labelled data and test data. Some conformal inference beyond exchangeability has attracted attention
+4
+
+(Tibshirani et al., 2019; Lei et al., 2021; Candès et al., 2021). In particular, Barber et al. (2022) proposed a
+general framework to implement conformal inference when the algorithms cannot treat data exchangeable and
+theoretically displayed the coverage deviations compared from exchangeability. However, how to decouple the
+dependence to achieve FCR control in the present framework remains a challenge.
+Taking a different but related perspective from multiple-testing, Bates et al. (2021) proposed a method to
+construct conformal p-values with data splitting and apply it to detect outliers with finite-sample FDR control.
+Zhang et al. (2022) extended that method and proposed a Jackknife implementation combined with automatic
+model selection. Jin and Candès (2022) considered a scenario that one aims to select some individuals of
+interest from the test sample and proposed a conformal p-value based method to control the FDR. Those
+existing works are not concerned about the construction of PIs, which differs with our focus essentially.
+1.3
+Organization and notations
+The remainder of this paper is organized as follows. We introduce the FCR-adjusted prediction and SCOP
+for valid FCR control in Section 2. Section 3 presents the theoretical properties of SCOP for ranking-based
+procedures. Numerical results and real-data examples are presented in Sections 4. Section 5 concludes the
+paper, and the technical proofs are relegated to the Supplementary Material.
+Notations. For a positive integer n, we use [n] to denote the index set {1, 2, ..., n}. Let A = {Ai : i = 1, ..., n}
+be a set of n real numbers, and S ⊆ [n] be an index subset. We use AS
+(ℓ) to denote the ℓ-th smallest value
+in {Ai : i ∈ S}. We use 1 {·} to denote the indicator function. For a real random sequences Xn and an
+non-negative real deterministic sequence an, we write Xn = Op(an) if for any ϵ > 0, there exists some constant
+C > 0 such that P(|Xn| > Can) ≤ ϵ. In our paper, the notations with subscript c or u refer to depending on
+the calibration set or the test set respectively.
+2
+Selective conditional conformal prediction
+Denote the index sets for the labelled data Dl and the test data Du as L = {1, . . . , 2n} and U = {2n +
+1, . . . , 2n + m}. The main prediction method studied in this paper is built upon the split conformal framework
+(Vovk et al., 2005; Lei et al., 2018), which is also called “inductive conformal prediction”. That is we randomly
+split Dl into two disjoint parts, the training set Dt and the calibration set Dc with n samples respectively. We
+can firstly train a prediction model ˆµ(X) on the Dl, and then compute the empirical quantiles of the residuals
+Ri = |Yi − ˆµ(Xi)| on the calibration set Dc. For Xj ∈ Du, the (1 − α)-marginal conformal PI is
+PIM
+j =
+�
+ˆµ(Xj) − QC(1 − α), ˆµ(Xj) + QC(1 − α)
+�
+,
+(2)
+where QC(1 − α) is the ⌈(1 − α)(n + 1)⌉-st smallest value in RC = {Ri = |Yi − ˆµ(Xi)| : i ∈ C}. Under the i.i.d.
+(or more generally, exchangeable) assumption on Dc ∪ {(Xj, Yj)}, the marginal PI in (2) enjoys the coverage
+guarantee, P
+�
+Yj /∈ PIM
+j
+�
+≤ α.
+5
+
+Suppose g : X → R be one plausible score function, which can be user-specified or estimated by the
+training data Dt. A particular selection procedure S can be applied to g(Xi) for i ∈ U to find the samples of
+interest. For simplicity, denote Ti = g(Xi) and those Xi’s with smaller values of Ti tend to be chosen. Denote
+the selected set as ˆSu = {i ∈ U : Ti ≤ ˆτ}, where ˆτ is the threshold. Different selective procedures S can be
+chosen from different perspectives, and we summarize the selection threshold ˆτ into three types.
+• (Fixed threshold) The ˆτ is user-specified or independent of the whole data. For example, ˆτ = t, where t
+is either known as a priori or could possibly be obtained from an independent process of Dc ∪ Du.
+• (Self-driven threshold) The ˆτ is only dependent on the scores {Ti : i ∈ U}. This type includes the
+Top-K which choose the first K individuals, and the quantile of Ti values in the test set which a given
+proportion of individuals with smallest Ti values in the test set, respectively (Fithian and Lei, 2020).
+• (Calibration-assisted selection) The ˆτ relies on the calibration set. For example, ˆτ is some quantile of
+true response Yi in calibration set, or the quantile based on both calibration and test set. In particular,
+one may employ some multiple testing procedures to achieve error rate control, such like FDR control
+based on the Benjamini–Hochberg (BH) procedure (Benjamini and Hochberg, 1995). Consequently, the
+{Ti : i ∈ C} is required to approximate the distribution of {Ti : i ∈ U}.
+Our goal is to construct conformal PIs for the selected subset ˆSu with the FCR control at α ∈ (0, 1).
+2.1
+Adjusted conformal prediction
+We firstly adapt the Benjamini and Yekutieli (2005)’s FCR-adjusted method to the present setting. Define
+Mj
+min := min
+y
+�
+| ˆSTj←y
+u
+| : j ∈ ˆSTj←y
+u
+�
+,
+where ˆSTj←y
+u
+denotes the selected subset when replacing Tj with value y. The FCR-adjusted conformal PIs
+are amount to marginally constructing larger 1 − α × Mj
+min/m PIs instead of 1 − α level in (2), i.e.,
+PIAD
+j
+=
+�
+ˆµ(Xj) − QC(1 − α × Mj
+min), ˆµ(Xj) + QC(1 − α × Mj
+min)
+�
+, j ∈ ˆSu.
+(3)
+Notice that given ˆµ(·) and ˆSu, PIAD
+j
+’s are not independent of each other because they all rely on the empirical
+quantile obtained from Dc, and therefore the proofs in Benjamini and Yekutieli (2005) are not readily extended
+to our setting. The following result demonstrates that the FCR-adjusted approach can successfully control
+the FCR for any selection threshold that is independent of the calibration set given training set.
+Proposition 2.1. Suppose that given Dt, {Ti : i ∈ C ∪ U} are independent random variables and the selection
+threshold ˆτ is independent of Dc. Then the FCR value of the FCR-adjusted method in (3) satisfies FCRAD ≤ α.
+For many plausible selection rules such as fixed-threshold selection, the Mj
+min can be replaced by the
+cardinality of the selected subset | �Su|. In practice, for ease of computation, one may prefer to use this
+6
+
+0.0
+0.1
+0.2
+0.0
+2.5
+5.0
+7.5
+10.0
+Ri
+density
+Marginal
+Conditional
+Test
+Figure 1: The densities of Ri for i ∈ Dc (in blue), Ri for i ∈ ˆSc (in green) and the density of Rj for j ∈ ˆSu (in red),
+respectively. There are 2n = 400 labeled data and m = 200 test data generated from a linear model with heterogeneous
+noise, where the details of the model are in Section 4.1. The selection rule is ˆS = {k : ˆµ(Xk) ≤ −1}.
+simplification, even though it does not have a theoretical guarantee for many data-dependent selection rules.
+The FCR-adjusted method is known to be quite conservative (Weinstein et al., 2013), because it does not
+incorporate the selection event into the calculation. Take the Top-K selection as an intuitive example. The
+selected set ˆSu is fixed with | ˆSu| = K and the FCR can be written as
+FCR = 1
+K
+�
+j∈U
+P
+�
+j ∈ ˆSu, Yj ̸∈ PIj
+�
+.
+(4)
+Since the marginal PIAD
+j
+reaches the 1 − αK/m confidence level for any fixed K, the FCR-adjusted method
+achieves the FCR control via
+FCRAD = 1
+K
+�
+j∈U
+P
+�
+j ∈ ˆSu, Yj ̸∈ PIAD
+j
+�
+≤ 1
+K
+�
+j∈U
+P
+�
+Yj ̸∈ PIAD
+j
+�
+≤ α,
+where the first inequality might be rather loose. A simple yet effective remedy is to use conditional calibration.
+2.2
+Selective conditional conformal prediction (SCOP)
+We start by making a decomposition of the FCR according to the contribution of each sample in the selected
+set ˆSu, given as P
+�
+Yj ̸∈ PIj |j ∈ ˆSu
+�
+P
+�
+j ∈ ˆSu
+�
+. Notice that the FCR can naturally be controlled at level α if
+the conditional control satisfies P(Yj ̸∈ PIj |j ∈ ˆSu) ≤ α, which sheds light on the construction of conditional
+conformal PI.
+In the regime of conformal inference, the conditional uncertainty of |Yj − ˆµ(Xj)| given j ∈ ˆSu can be reliably
+approximated using the calibration set Dc, enabling us to construct model/distribution-agnostic conditional
+7
+
+PIs. To be specific, we conduct the selective algorithm S on the fitted score values {Ti = g(Xi) : i ∈ C}
+and obtain the post-selection calibration set ˆSc = {i ∈ C : Ti ≤ ˆτ} with the same threshold ˆτ. Notice that
+ˆSc is formed via the same selection criterion with ˆSu, and thus we utilize the residuals Ri for i ∈ ˆSc to
+approximately characterize the conditional uncertainty of Rj for j ∈ ˆSu. To visualize the effect, we consider
+a linear model with heterogeneous noise, where we use ordinary least-squares for predictions and select
+ˆSu = {j ∈ U : ˆµ(Xj) ≤ −1}. In Figure 1, we display the densities of Ri for i ∈ Dc, Ri for i ∈ ˆSc and
+the density of Rj for j ∈ ˆSu, respectively. The selection procedure significantly distorts the distribution of
+residuals, but the conditional uncertainty on ˆSu can be well approximated by that on ˆSc.
+The conditional conformal PI for j ∈ ˆSu can accordingly be constructed as
+PISCOP
+j
+=
+�
+ˆµ(Xj) − Q
+ˆ
+Sc(1 − α), ˆµ(Xj) + Q
+ˆ
+Sc(1 − α)
+�
+,
+(5)
+where Q
+ˆ
+Sc(1 − α) is the ⌈(1 − α)(| ˆSc| + 1)⌉-st smallest value in R
+ˆ
+Sc = {Ri : i ∈ ˆSc}. We refer this procedure
+as Selective COnditional conformal Prediction (SCOP) and summarize it in Algorithm 1.
+The following theorem shows that SCOP can control the FCR at α for exchangeable selective procedures.
+Further, if the selection scores Ti are continuous (or almost surely distinct), we can obtain a lower bound for
+the FCR value, guaranteeing that the SCOP is nearly exact in O(n−1).
+Theorem 1. Suppose {Ti : i ∈ C ∪ U} are exchangeable random variables, and the threshold ˆτ is also
+exchangeable with respective to the {Ti : i ∈ C ∪ U}. For each j ∈ U, the conditional miscoverage probability is
+bounded by
+P
+�
+Yj ̸∈ PISCOP
+j
+|j ∈ ˆSu
+�
+≤ α.
+(6)
+Further, the FCR value of the SCOP algorithm is controlled at FCRSCOP ≤ α. In addition, if Ti follows a
+continuous distribution for i ∈ C ∪ U and P(| ˆSu| > 0) = 1, we also have
+P
+�
+Yj ̸∈ PISCOP
+j
+|j ∈ ˆSu
+�
+≥ α −
+1
+n + 1
+and FCRSCOP ≥ α −
+1
+n+1.
+Under the exchangeable assumption, the FCR results actually match the marginal miscoverage results of
+original conformal PIs (Vovk et al., 2005). This theorem relies on exchangeability in two ways, i.e., the fitted
+selection score {Ti : i ∈ C ∪ U} are exchangeable and the selection threshold ˆτ is assumed to keep the same
+value by swapping Tj and Tk for any j, k ∈ C ∪ U. The former one is commonly used in conformal inference
+and holds easily when the data are i.i.d given Dt (Lei et al., 2018). The later one imposes restrictions on
+the selection procedures and can be fulfilled with some practical thresholds. The simplest case is the fixed
+threshold. Another popular example is that ˆτ is some quantile of {Ti : i ∈ C ∪ U}. However, many selection
+procedures may be excluded, such as the Top-K selection. In such cases, the threshold ˆτ is only determined
+by the test data U, which does not treat the data points from calibration and test sets symmetrically. We will
+next explore the effectiveness of the proposed SCOP for more general selection procedures.
+8
+
+Algorithm 1 Selective COnditional conformal Prediction (SCOP)
+Input: Labeled set Dl, test set Du, selection procedure S, target FCR level α ∈ (0, 1).
+Step 1 (Splitting and training) Split Dl into training set Dt and calibration set Dc with equal size n. Fit
+prediction model ˆµ(·) and score function g (if needed) on the training set Dt.
+Step 2 (Selection) Compute the scores: TC = {Ti = g(Xi) : i ∈ C} and TU = {Ti = g(Xi) : i ∈ U}. Apply
+the selective procedure S to TC ∪ TU and obtain the threshold value ˆτ. Obtain the post-selection subsets:
+ˆSu = {i ∈ U : Ti ≤ ˆτ} and ˆSc = {i ∈ C : Ti ≤ ˆτ}.
+Step 3 (Calibration) Compute residuals: RSc = {Ri = |Yi − ˆµ(Xi)| : i ∈ ˆSc}. Find the ⌈(1−α)(| ˆSc|+1)⌉-st
+smallest value of RSc, Q
+ˆ
+Sc(1 − α).
+Step 4 (Construction) Construct PI for each j ∈ ˆSu as PISCOP
+j
+= [ˆµ(Xj)−Q
+ˆ
+Sc(1−α), ˆµ(Xj) +Q
+ˆ
+Sc(1−α)].
+Output: Prediction intervals {PISCOP
+j
+: j ∈ ˆSu}.
+Remark 2.1. In predictive inference, several works considered to approximately construct the conditional PI
+(Chernozhukov et al., 2021; Feldman et al., 2021), i.e.,
+P (Yj /∈ PI(Xj)|Xj = x) ≤ α,
+for any x ∈ X.
+(7)
+However, it is well known that achieving “fully" conditional validity in (7) is impossible in distribution-free
+regime (Lei et al., 2013; Foygel Barber et al., 2020). Our conditional miscoverage control in (6) is a weaker
+guarantee compared with (7), since we only consider the selection events. For more discussion about these two
+conditional guarantees, we refer to Appendix B in Weinstein and Ramdas (2020). The SCOP can leverage
+the post-selection calibration set to approximate the selective conditional distribution of residuals, which
+contributes to achieve a better conditional coverage. In addition, the conditional calibration of SCOP provides
+an anti-conservative lower bound for FCR value in the continuous case.
+3
+Ranking-based selection
+In this section, we consider a general class of selection procedures named ranking-based selection and discuss
+the conditions under which the proposed SCOP is able to control the FCR. In Sections 3.1 and 3.2, we discuss
+the FCR control for the self-driven selection procedures and calibration-assisted ones, respectively. Then, in
+Section 3.3, we demonstrate the effectiveness of the SCOP procedure when the selection procedures based on
+conformal p-values are used.
+We begin with some general assumptions and notations. For simplicity, we suppose Ti ∈ [0, 1] and the
+selection algorithm S conducted on {Ti : i ∈ C ∪ U} outputs a ranking threshold ˆκ ∈ [m]. Say, we have the
+selection threshold ˆτ = TU
+(ˆκ) as the ˆκ-th smallest value in TU = {Tj : j ∈ U}. Then the selected subset of the
+9
+
+test set can be rewritten as
+ˆSu =
+�
+j ∈ U : Tj ≤ TU
+(ˆκ)
+�
+.
+(8)
+The ranking-based procedure in (8) incorporates many practical examples, such as Top-K selection, quantile-
+based selection, step-up procedures (Fithian and Lei, 2020) and the well-known BH procedure1 (Benjamini
+and Hochberg, 1995).
+With the ranking based selection, we have | ˆSu| = ˆκ, which is usually random and coupled to each test
+sample Xj ∈ Du. To decouple the dependence, we introduce Lemma 1 to control the FCR through conditioning
+on the leave-one-out data set Du,−j, which is the test set Du without the sample j. Denote EDu,−j[·] and
+PDu,−j(·) as the conditional expectation and probability given Du,−j. Let ˆκj←tu be the ranking threshold
+obtained from the selection algorithm S by replacing Tj with some deterministic value tu ∈ [0, 1].
+Lemma 1. Suppose | ˆSu| > 0 almost surely and ˆκ = ˆκj←tu holds for any j ∈ ˆSu. If the conditional false
+coverage probability satisfies
+���PDu,−j
+�
+Yj ̸∈ PIj
+��j ∈ ˆSu
+�
+− α
+��� ≤ ∆(Du,−j),
+(9)
+where ∆(Du,−j) only depends on the data set Du,−j, then we have
+| FCR −α| ≤ E
+�
+� 1
+| ˆSu|
+�
+j∈ ˆ
+Su
+∆(Du,−j)
+�
+� .
+The leave-one-out technique often appears in the literature about FDR control under dependence (Heesen
+and Janssen, 2015; Fithian and Lei, 2020; Luo et al., 2022). The key is to decompose the FCR into the
+summation of conditional miscoverage probability of each candidate given other test samples, i.e,
+FCR = E
+�
+��
+j∈U
+1
+ˆκj←tu PDu,−j
+�
+Yj ̸∈ PIj |j ∈ ˆSu
+�
+PDu,−j
+�
+j ∈ ˆSu
+�
+�
+� .
+We may regard the term ∆(Du,−j) in (9) as the individual FCR contribution of the j-th candidate in ˆSu. The
+detailed proof of Lemma 1 is deferred to Appendix C.1.
+Next, we introduce two universal assumptions to find how the conditional false coverage probability in
+(9) holds and further control the FCR of SCOP with the ranking-based selection. Denote the cumulative
+distribution functions (CDF) of Ri and Ti as FT (·) and FR(·), respectively. Let F(R,T )(·, ·) be the joint CDF
+of (Ri, Ti).
+Assumption 1. The score function g depends only on the training set. Suppose {Ti : i ∈ C ∪ U} and
+{Ri : i ∈ C ∪ U} are both i.i.d. continuous random variables. There exists some ρ ∈ (0, 1) such that
+d
+drF(R,T )
+�
+F −1
+R (r), F −1
+T (t)
+�
+≥ ρt,
+holds for any t ∈ (0, 1) and r ∈ (0, 1).
+1The BH procedure is also an example of step-up procedures, see Fithian and Lei (2020)
+10
+
+Assumption 2. There exists some deterministic value tu ∈ [0, 1] such that ˆκj←tu = ˆκ holds for any j ∈ ˆSu,
+and ˆκj←tu ≤ ˆκ + Iu holds for any j ∈ U \ ˆSu and some positive integer Iu ≤ m.
+To facilitate our technical development, we impose mild distributional assumption on the joint CDF of
+(Ri, Ti) in Assumption 1. It is worth noticing that the same selected set ˆSu and ˆSc can be obtained if one
+applies the ranking-based selection procedure to the transformed scores {FT (Ti) : i ∈ U} instead of the scores
+{Ti : i ∈ U}. Also, the transformed residuals {FR(Ri) : i ∈ C ∪ U} keep the original order as the residuals
+{Ri : i ∈ C ∪ U} in the conformal coverage control. Therefore, without loss of generality, we can assume that
+Ti
+i.i.d.
+∼ Unif([0, 1]) and Ri
+i.i.d.
+∼ Unif([0, 1]) for i ∈ C ∪ U in the theoretical analysis. Then the condition on
+CDF in Assumption 1 will reduce to
+d
+drF(R,T )(r, t) ≥ ρt which appears quite weak.
+For Assumption 2, we can verify that ˆκ = ˆκj←tu under the event {j ∈ ˆSu} in many cases, such as
+the quantile-based selection procedure and BH procedure. For the selection procedures with fixed ranking
+threshold, such as quantile-based selection and Top-K selection, Assumption 2 is clearly satisfied with tu = 0
+and Iu = 0. For the BH procedure based on the conformal p-values, taking tu as 0 for each j ∈ ˆSu leads a
+smaller p-value pj. By the property of the BH procedure, for j ∈ ˆSu, assigning pj to a smaller value will not
+change the rejection set (Fithian and Lei, 2020), and hence we have κj←0 = ˆκ for j ∈ ˆSu. In Section 3.3, we
+also show that Iu = Op(log m) for the BH procedure. From now on, we write ˆκ(j) = ˆκj←tu for simplicity.
+3.1
+FCR control for self-driven selection
+When the self-driven selection procedures are used, the samples in the selected calibration set ˆSc and the
+selected test set ˆSu are not exchangeable, but the original samples from the calibration set and the test set
+are exchangeable. The following theorem provides delicate bounds for the conditional miscoverage gap of the
+SCOP.
+Theorem 2. Under Assumptions 1 and 2, for any absolute constant C ≥ 1, if 8C log n/(nTU\{j}
+(ˆκ(j)) ) ≤ 1 holds
+almost surely, the conditional miscoverage probability can be bounded by
+PDu,−j
+�
+Yj ̸∈ PISCOP
+j
+���j ∈ ˆSu
+�
+≤ α + ∆(Du,−j),
+and
+PDu,−j
+�
+Yj ̸∈ PISCOP
+j
+���j ∈ ˆSu
+�
+≥ α − 2∆(Du,−j),
+where
+∆(Du,−j) =
+8C log n
+ρTU\{j}
+(ˆκ(j)−Iu)
+�
+�6C log n
+nTU\{j}
+(ˆκ(j))
++
+TU\{j}
+(ˆκ(j)) − TU\{j}
+(ˆκ(j)−1)
+TU\{j}
+(ˆκ(j))
+�
+� +
+2
+�
+TU\{j}
+(ˆκ(j)) − TU\{j}
+(ˆκ(j)−Iu)
+�
+TU\{j}
+(ˆκ(j)−Iu)
+.
+(10)
+Our theorem is closely connected to Barber et al. (2022)’s Theorem 2a. Both theorems involve assessing
+how the deviations from the “idealized” exchangability would affect the actual miscoverage level. However,
+11
+
+the interpretations are very different. Barber et al. (2022) showed that under assumption that the test and
+calibration samples are non-exchangeable, the miscoverage gap can be bounded by an error term regarding the
+total variation between the two samples. Whereas in SCOP the deviation comes from the possible violation of
+the similarity between the distributions of {Ri : i ∈ ˆSc} and {Rj : j ∈ ˆSu}.
+Remark 3.1. The technical difficulty in proving Theorem 2 lies in coping with the dependence of ˆSc
+and ˆSu. To address this problem, we introduce virtual post-selection test set and calibration set, ˆS(j)
+u
+=
+�
+i ∈ U : Ti ≤ TU\{j}
+(ˆκ(j))
+�
+and ˆS(j)
+c
+=
+�
+i ∈ C : Ti ≤ TU\{j}
+(ˆκ(j))
+�
+respectively. We denote the corresponding virtual
+conformal PI constructed by ˆS(j)
+c
+as PIj( ˆS(j)
+c ). For clarity, we rewrite the real conformal PI constructed by ˆSc in
+Algorithm 1 as PIj( ˆSc) ≡ PISCOP
+j
+. Notice that, the threshold TU\{j}
+(ˆκ(j)) and the virtual selected calibration set ˆS(j)
+c
+are independent of the test candidate j. Therefore, the test candidate j and the calibration candidate k are ex-
+changeable in the set ˆS(j)
+c
+∪{j} under the selection conditions. It remains to control two conditional miscoverage
+gaps: PDu,−j
+�
+j ̸∈ PIj( ˆS(j)
+c )|j ∈ ˆS(j)
+u
+�
+− α and PDu,−j
+�
+j ̸∈ PIj( ˆSc)|j ∈ ˆSu
+�
+− PDu,−j
+�
+j ̸∈ PIj( ˆS(j)
+c )|j ∈ ˆS(j)
+u
+�
+;
+the former can be bounded as in conventional conformal inference.
+Our theorem shows that a tight control of the deviation term ∆(Du,−j) in (10) leads to effective FCR
+control. Next we carefully interpret the bound and present more explicit settings in which the FCR achieves
+or is very close to the nominal level. Observe that controlling ∆(Du,−j) actually boils down to establishing
+the upper bound of the difference TU\{j}
+(ˆκ(j)) − TU\{j}
+(ˆκ(j)−Iu) and the lower bound of the denominator TU\{j}
+(ˆκ(j)−Iu). To
+guarantee that the denominator will stay away from 0, we impose the following assumption on ˆκ.
+Assumption 3. The ranking threshold satisfies ˆκ ≥ γm for some γ ∈ (0, 1).
+The lower bound on ˆκ in Assumption 3 is mild and reasonable, since the FCR control will be extremely
+difficult when | ˆSu|/n = ˆκ/n = o(1) for a small level α. Applying the well-known representation of spacing
+between consecutive order statistics (c.f. Lemma D.1) to {Ti}i∈U\{j}, together with Assumption 3, we can
+obtain the following FCR control result for self-driven selection procedures.
+Theorem 3. Under Assumptions 1-3. If γ > Iu/m, the FCR value of SCOP with self-driven selection
+procedures can be controlled at
+FCRSCOP = α + O
+� log2(n ∨ m)
+ργ(γ − Iu/m)
+�Iu
+m + 1
+n
+��
+.
+In the asymptotic regime, FCRSCOP is exact if Iu = o(m), that is lim(n,m)→∞ FCRSCOP = α. Recalling
+that for quantile-based selection and Top-K selection, we have Iu = 0, and thus Theorem 3 guarantees the
+FCR of SCOP with such selection procedures can attain the target level in a nearly optimal rate (up to a
+logarithmic factor).
+12
+
+3.2
+FCR control for calibration-assisted selection
+For calibration-assisted selective procedures, the analysis is more complex because the ranking threshold ˆκ
+depends also on the calibration set Dc. It implies that a more tractable ranking threshold is needed to decouple
+the dependence on the selected samples and the calibration samples simultaneously. That is for any j ∈ ˆSu and
+k ∈ C, let ˆκ(j,k) ≡ ˆκj←tu,k←tc be the ranking threshold by replacing Tj with tu and Tk with tc simultaneously.
+The virtual post-selection calibration set is further defined as ˆS(j,k)
+c
+=
+�
+i ∈ C \ {k} : Ti ≤ TU\{j}
+(ˆκ(j,k))
+�
+.
+The following assumption, an analog of Assumption 2, is imposed to restrict the change in the ranking
+threshold after replacing one calibration score.
+Assumption 4. There exists some tc ∈ R and some positive integer Ic ≤ m such that ˆκ ≤ ˆκk←tc ≤ ˆκ + Ic
+holds for any k ∈ C.
+The following theorem is parallel with Theorem 2.
+Theorem 4. Under Assumptions 1-4, for the calibration-assisted selection, the conditional miscoverage
+probability of SCOP satisfies
+PDu,−j
+�
+Yj ̸∈ PISCOP
+j
+��j ∈ ˆSu
+�
+≤ α + EDu,−j
+�
+max
+k
+∆(D(j,k))
+�
+,
+and
+PDu,−j
+�
+Yj ̸∈ PISCOP
+j
+��j ∈ ˆSu
+�
+≥ α − 2EDu,−j
+�
+max
+k
+∆(D(j,k))
+�
+,
+where
+∆(D(j,k)) :=
+2TU\{j}
+(ˆκ(j,k))
+�
+R
+ˆ
+S(j,k)
+c
+(U (j,k)) − R
+ˆ
+S(j,k)
+c
+(L(j,k))
+�
+�
+TU\{j}
+(ˆκ(j,k)−Iu−Ic)
+�2
++
+4
+�
+TU\{j}
+(ˆκ(j,k)) − TU\{j}
+(ˆκ(j,k)−Iu−Ic)
+�
+TU\{j}
+(ˆκ(j))
++
+d(j,k)
+| ˆS(j,k)
+c
+| + 1
+,
+(11)
+with d(j,k) = �
+i∈ ˆ
+S(j,k)
+c
+1
+�
+TU\{j}
+(ˆκ(j,k)−Ic−1) < Ti ≤ TU\{j}
+(ˆκ(j,k))
+�
+, U (j,k) = ⌈(1 − α)(| ˆS(j,k)
+c
+| + 2)⌉ + d(j,k) and L(j,k) =
+⌈(1 − α)(| ˆS(j,k)
+c
+| + 2 − d(j,k))⌉ − 2.
+Remark 3.2. We can see that all the terms in (11) are independent of the samples j ∈ ˆSu and k ∈ C. The
+quantity d(j,k) measures the size difference between the real calibration set ˆSc and the virtual calibration set
+ˆS(j,k)
+c
+. The term R
+ˆ
+S(j,k)
+c
+(U (j,k)) − R
+ˆ
+S(j,k)
+c
+(L(j,k)) represents the largest possible distance of the corresponding quantiles
+in ˆSc and ˆS(j,k)
+c
+, which can be bounded in ˆS(j,k)
+c
+conditional on the data set Du,−j. The remaining parts in
+maxk ∆(D(j,k)) rely on the difference between thresholds from TU\{j}.
+Equipped with the conditional miscoverage gap in Theorem 4, we can obtain the FCR control results of
+SCOP with calibration-assisted selection in the following theorem.
+Theorem 5. Under Assumptions 1-4. If γ > (Ic + Iu)/m, the FCR value of SCOP for calibration-assisted
+selection can be controlled at
+FCRSCOP = α + O
+�
+log2(n ∨ m)
+ρ(γ − (Ic + Iu)/m)2
+�Ic + Iu
+m
++ 1
+n
+��
+.
+13
+
+Similar to the results with self-driven selection, the SCOP can control the FCR around the target value
+with a small gap. To decouple the dependence between the ranking threshold and calibration set, an addition
+term Ic/m appears in Theorem 5, regarding to the effect of replacing one calibration sample. If Ic ∨Iu = o(m),
+then we can take m = exp{o(n
+1
+2 )} and have lim(n,m)→∞ FCRSCOP = α.
+3.3
+Prediction-oriented selection with conformal p-values
+We discuss the implementation of the SCOP with a special calibration-assisted selection procedure, the
+selection via multiple testing based on conformal p-values. The concept of the conformal p-value was proposed
+by Vovk et al. (2005). Similar to the conformal PI, the conformal p-values enjoy model/distribution-free
+properties. Recently, there exists some works to apply conformal p-values to implement sample selection from
+a multiple-testing perspective, such as Bates et al. (2021) and Jin and Candès (2022).
+In particular, Jin and Candès (2022) investigated the prediction-oriented selection problem, aiming to
+select samples whose unobserved outcomes exceed some specified values while control the proportion of falsely
+selected units. This problem can be viewed as the following multiple hypothesis tests: for i ∈ U and some
+b0 ∈ R,
+H0,i : Yi ≥ b0
+v.s.
+H1,i : Yi < b0.
+By choosing a monotone function g0 : R+ → [0, 1], one could take the score function as g(x) = g0(ˆµ(x) − b0)
+and compute the conformity scores as {Ti = g(Xi) : i ∈ C ∪ U}.
+Denote the null set of calibration samples as C0 = {i ∈ C : Yi ≥ b0} and its size as n0 = |C0|. Given the
+conformity scores {Ti : i ∈ C0} in the calibration set, the conformal p-value for each test data point can be
+calculated by2
+pj := p(Xj) = 1 + |{i ∈ C0 : Ti ≤ Tj}|
+n0 + 1
+,
+for j ∈ U.
+(12)
+To control FDR at the target level β ∈ (0, 1), we may deploy BH procedure to {pj : j ∈ U} and obtain the
+rejection set ˆSu.
+Proposition 3.1. Let pU
+(1) ≤ ... ≤ pU
+(m) be order statistics of conformal p-values in the test set U. For any
+i ∈ U, it holds that {pi ≤ pU
+(ˆκ)} = {Ti ≤ TU
+(ˆκ)}.
+Proposition 3.1 indicates that using the conformal p-values in (12) to obtain ˆSu is equivalent to using the
+conformity scores in TU with the same ranking threshold ˆκ, that is ˆSu = {i ∈ U : pi ≤ pU
+(ˆκ)} ≡ {i ∈ U : Ti ≤
+TU
+(ˆκ)}. Further, we can also obtain the post-selection calibration set by ˆSc = {i ∈ C : Ti ≤ TU
+(ˆκ)}. Therefore,
+we can frame the BH procedures with conformal p-values as a calibration-assisted selection in Section 3.2.
+2Under our continuous assumption, we present the form of conformal p-value without ties in the conformity scores. For the
+tie-breaking form, please refer to Bates et al. (2021) for more details.
+14
+
+Algorithm 2 SCOP under selection with conformal p-values
+Input: Training data Dt, calibration data Dc, test data Du, threshold sequence {δ(r) : r ∈ [m]}.
+Step 1 Fit prediction model ˆµ(·) and score function g(·) on Dt. Compute the score values TC = {Ti =
+g(Xi) : i ∈ C} and TU = {Ti = g(Xi) : i ∈ U}.
+Step 2 Compute the conformal p-values {pi : i ∈ U} according to (12) based on DC0. Apply the BH
+procedure with target level β to TU and obtain (13). Obtain the post-selection subsets: ˆSu = {i ∈ U : Ti ≤
+TU
+(ˆκ)} and ˆSc = {i ∈ C : Ti ≤ TU
+(ˆκ)}.
+Step 3 Compute residuals: RSc = {Ri = |Yi − ˆµ(Xi)| : i ∈ ˆSc}. Find the ⌈(1 − α)(| ˆSc| + 1)⌉-st smallest
+value of RSc, denoted by Q
+ˆ
+Sc(1 − α).
+Step 4 Construct PI for each j ∈ ˆSu as PIj = [ˆµ(Xj) − Q
+ˆ
+Sc(1 − α), ˆµ(Xj) + Q
+ˆ
+Sc(1 − α)].
+Output: {PIj : j ∈ ˆSu}.
+To study the FCR control with selection procedures based on conformal p-values, we consider a more
+general class of step-up procedures introduced by Fithian and Lei (2020). Let 0 ≤ δ(1) ≤ · · · ≤ δ(m) ≤ 1
+denote an increasing sequence of thresholds, we choose the ranking threshold for step-up procedures as
+ˆκ = max
+�
+r : pU
+(r) ≤ δ(r)
+�
+,
+(13)
+where pU
+(r) is the rth-smallest conformal p-value. Specially, the BH procedure takes δ(r) = rβ/m. We
+summarize the SCOP with the step-up selection procedures in Algorithm 2.
+To adapt Assumptions 2 and 4, we can simply take ˆκ(j) = ˆκj←0 and ˆκ(k) = ˆκk←1 by replacing Tj with 0
+for j ∈ U and Tk with 1 for k ∈ C, respectively. From Lemma 1 in Fithian and Lei (2020), we have ˆκ(j) = ˆκ
+for any j ∈ ˆSu in the step-up procedures. The next proposition characterizes the magnitudes of ˆκ(j) − ˆκ for
+any j ̸∈ ˆSu and ˆκ(k) − ˆκ for any k ∈ C.
+Proposition 3.2. Suppose {Xi : i ∈ C ∪ U} are i.i.d. continuous random variables. Let Ω(r) = {ℓ ∈ [n0] :
+δ(r) <
+ℓ+1
+n0+1 ≤ δ(r + 1)}. For step-up procedures (13) using conformal p-values defined in (12) and any
+absolute constant C > 1,
+1. For any j ∈ ˆSu, we have ˆκ(j) = ˆκ. In addition, for any j ∈ U \ ˆSu,
+ˆκ(j) − ˆκ ≤ 12C log m + 8Cm log m
+n0 + 1
+max
+⌈γm⌉<r≤m−1 |Ω(r)|,
+holds with probability at least 1 − 2m−C+1.
+2. For any k ∈ C, we have ˆκ(k) ≥ ˆκ and
+ˆκ(k) − ˆκ ≤ 12C log m + 8Cm log m
+n0 + 1
+,
+holds with probability at least 1 − 2m−C+1.
+15
+
+Remark 3.3. In fact, the size of the set Ω(r) quantifies the sensitivity of the step-up procedure to the order
+changes in conformal p-values. Notice that, if we set the value of Tj to 0, the ranks of test scores in the set
+{i ∈ U \ {j} : pi ≤ pj} will increase 1. In accordance with the definition of the ranking threshold in (13), we
+know that ˆκ(j) = max
+�
+ˆκ < r ≤ m : δ(r) < pU
+(r) ≤ δ(r + 1), pU
+(r) ≤ pj
+�
+. By (12), each conformal p-value can
+only take values in { ℓ+1
+n0+1 : ℓ ∈ [n0]}. Consequently, the gap ˆκ(j) − ˆκ tends to be larger when |Ω(r)| increases.
+For BH procedure with δ(r) = βr/m, it holds that max1≤r≤m−1 |Ω(r)| ≤ ⌈ (n+1)β
+m+1 ⌉. Therefore, by the
+definitions of Iu and Ic in Assumptions 2 and 4 respectively, we can guarantee that Iu/m = Op( log(n∨m)
+n∧m
+) and
+Ic/m = Op( log(n∨m)
+n∧m
+). Invoking the FCR bound in Theorem 5, we have the following FCR control results for
+SCOP with the step-up procedures.
+Theorem 6. Suppose {Xi : i ∈ C ∪ U} are i.i.d. continuous random variables and the joint CDF of (Ri, Ti)
+satisfies Assumption 1. If the ranking threshold is output by the step-up procedures (including the BH method)
+based on conformal p-values, and max1≤r≤m−1 |Ω(r)| = O(1) holds, then the FCR value of SCOP can be
+controlled at
+FCRSCOP = α + O
+�log3(n ∨ m)
+n ∧ m
+�
+.
+Theorem 3.5 isn’t trivially extended from Theorem 5. Note that ˆκ in (13) is partial calibration-assisted
+threshold since the conformal p-values pj in (12) are only dependent on null samples in calibration set, i.e,
+DC0. That results in ˆSc contains both the null and alternative samples and both have different dependence
+structures between ˆSu. Theorem 3.5 could achieve asymototic FCR control if m = exp{o(n
+1
+3 )}.
+4
+Numerical results
+4.1
+Simulation studies
+We illustrate the breadth of applicability of the proposed methods via comprehensive simulation studies. In
+each setting, we generate i.i.d. 10-dimensional covariates from Xi ∼ Unif([−1, 1])10 and the corresponding
+responses as Yi = µ(Xi) + ϵi. Three data generating scenarios are considered with different configurations of
+µ(·) and distributions of ϵi’s.
+• Scenario A (Linear model with heterogeneous noise): µ(X) = X⊤β, where β is randomly sampled
+from Unif([−1, 1])10. The noise is heterogeneous which follows ϵ|X ∼ N(0, 1 + |µ(X)|).
+• Scenario B (Nonlinear model): µ(X) = X(1)X(2) + X(3) − 2 exp(X(4) + 1), where X(k) denotes the
+k-th element of vector X. The noise is from ϵ ∼ N(0, 1) and is independent of covariates X.
+• Scenario C (Aggregation model): µ(X) = 4(X(1) + 1)|X(3)|1X(2)>−0.4 + 4(X(1) − 1)1X(2)≤−0.4.The
+noise is ϵ ∼ N(0, 1) and independent of X too.
+16
+
+We fix the labeled data size 2n = 400 at first and equally split it into Dt and Dc. The regression model ˆµ(·)
+is fitted on Dt by ordinary least-squares (OLS) for Scenario A, support vector machine (SVM) for Scenario B
+and random forest (RF) for Scenario C, respectively. The SVM and RF are implemented by R packages ksvm
+and randomForest with default parameters. In most cases, the selection score is chosen as the same as the
+prediction value, i.e., Ti = ˆµ(Xi).
+We want to select one subset via ˆSu = {i ∈ U : Ti ≤ ˆτ}, where ˆτ is the threshold. To illustrate the wide
+applicability of the proposed method, several selection thresholds ˆτ are considered.
+(1) T-cal(q): q%-quantile of true response Y in calibration set, that is ˆτ is q%-quantile of {Yi : i ∈ Dc};
+(2) T-test(q): q%-quantile of predicted response ˆµ(X) in test set, i.e. ˆτ = TU
+(qm/100);
+(3) T-exch(q): q%-quantile of predicted response ˆµ(X) in both calibration set and test set, that is ˆτ is the
+q%-quantile of TC∪U = {Ti : Ti ∈ Dc ∪ Du};
+(4) T-cons(b0): a pre-determined constant value b0, i.e., ˆτ = b0;
+(5) T-pos(b0,β): The prediction-oriented selection proposed by Jin and Candès (2022), where one would
+like to select those test samples with response Y smaller than b0 while controlling the FDR level at
+β = 0.2. Here the threshold ˆτ is computed by the BH procedure with conformal p-values in Section 4.
+(6) T-top(K): Kth smallest value of predicted responses ˆµ(X) in test set, i.e. ˆτ = TU
+(K).
+(7) T-clu: One popular choice of thresholds is based on clustering, where the boundary value of two
+possible groups is a natural cut point. To be specific, we want to find the threshold ˆτ that minimizes
+the within-group sum of squares, i.e,
+ˆτ = arg min
+t∈TC∪U
+�
+i∈S1(t)
+(Ti − ¯T
+S1(t))2 +
+�
+k∈Sc
+1(t)
+(Tk − ¯T
+Sc
+1(t))2,
+where S1(t) = {i ∈ C ∪ U : Ti ≤ t}, Sc
+1(t) = C ∪ U/S1(t) and ¯T
+S1(t) is the sample mean in S1(t).
+Among all the considered threshold selections, only the T-exch(q), T-cons(b0) and T-clu satisfy the
+exchangeability with respective to {Ti : i ∈ C ∪ U}. The threshold T-top(K) is actually a special case of
+T-test(q) with K = [qm/100]. We apply SCOP to construct PIs for the selected individuals in ˆSu with
+target FCR level α = 10%. Two benchmarks are included for comparison. One is to directly construct a
+(1 − α)-marginal prediction interval as (2) for each selected sample based on the whole calibration set. We
+refer this method as ordinary conformal prediction (OCP) and notice that it takes no account of the selection
+effects. Another one is the FCR-adjusted conformal prediction (ACP), which builds 1 − α| ˆSu|/m level PI as
+(3) for each selected individual. The performances are compared in terms of the FCR and average length (AL)
+of constructed PIs among 1,000 repetitions.
+17
+
+method
+SCOP
+OCP
+ACP
+T−cal
+T−test
+T−exch
+Scenario A
+Scenario B
+Scenario C
+30
+60
+90
+30
+60
+90
+30
+60
+90
+5
+10
+15
+20
+25
+5
+10
+15
+20
+5
+10
+15
+q%
+FCR(%)
+T−cal
+T−test
+T−exch
+Scenario A
+Scenario B
+Scenario C
+30
+60
+90
+30
+60
+90
+30
+60
+90
+10.0
+12.5
+15.0
+17.5
+5.0
+5.5
+6.0
+6.5
+7.0
+7.5
+5
+6
+7
+8
+q%
+Length
+Figure 2: Empirical FCR (%) and average PI length for quantile based thresholds with varying quantile level q. The
+black dashed line represents the target FCR level 10%.
+We firstly fix the size of test data Du as m = 200 and consider the three quantile-based thresholds:
+T-cal(q), T-test(q) and T-exch(q). Figure 2 displays the estimated FCR and AL of PIs through varying
+the quantile level q% from 20% to 100%. Across all the settings, it is evident that the SCOP is able to deliver
+quite accurate FCR control and have more narrowed PIs. As expected, the OCP yields the same lengths of
+PIs under all the settings and can only control the FCR under q% = 100%, that is the situation all the test
+data are included without selection. This can be understood since the OCP builds the marginal PIs using the
+whole calibration set without consideration of the selection procedure and thus possesses the length of PI as
+2QC(1 − α) in (2). The ACP results in considerably conservative FCR levels and accordingly it performs not
+well in terms of the AL of PIs in most settings. This is not surprising as the ACP marginally constructs much
+larger 1 − α| ˆSu|/m PIs to ensure the FCR control than the target level.
+In Table 1, we present the results of the remaining four thresholds, including T-cons(b0), T-pos(b0,β),
+T-top(K) and T-clu. Here, we fix the constant b0 for both T-cons(b0) and T-pos(b0,β) as the 30%-quantile
+of the true response Yi’s, and choose the target FDR level β = 20% for T-pos(b0,β) and K = 60 for T-top(K).
+It can be seen that the FCR levels of SCOP are close to the nominal level. The SCOP also achieves satisfactory
+narrowed PIs under all the scenarios. The OCP leads to much different FCR levels but same average length
+18
+
+Table 1: Comparisons of Empirical FCR (%) and average length (AL) under different scenarios and thresholds
+with target FCR α = 10%. The sample sizes of the calibration set and the test set are fixed as n = m = 200.
+T-con(b0)
+T-pos(b0, 20%)
+T-top(60)
+T-clu
+SCOP
+OCP
+ACP
+SCOP
+OCP
+ACP
+SCOP
+OCP
+ACP
+SCOP
+OCP
+ACP
+Scenario A
+FCR
+10.02
+14.27
+4.87
+7.31
+13.56
+2.51
+9.75
+15.53
+4.85
+9.78
+10.16
+5.07
+AL
+11.77
+9.91
+14.77
+15.52
+9.91
+22.53
+12.02
+9.91
+15.02
+10.15
+9.91
+12.86
+Scenario B
+FCR
+9.77
+17.41
+7.24
+9.61
+16.33
+7.28
+9.63
+17.70
+6.86
+9.75
+14.98
+7.41
+AL
+5.86
+4.70
+6.43
+5.73
+4.70
+6.25
+5.95
+4.70
+6.53
+11.88
+4.70
+5.99
+Scenario C
+FCR
+9.74
+13.07
+3.62
+9.70
+12.06
+3.88
+10.03
+12.77
+3.92
+9.89
+12.66
+3.67
+AL
+5.82
+5.27
+7.41
+5.72
+5.27
+7.19
+5.73
+5.27
+7.23
+5.77
+5.27
+7.33
+of PIs by different selection thresholds. This corroborates the insight that the OCP is unable to give a valid
+coverage guarantee on the selected ones. In contrast, the FCRs of ACP are overly conservative and in turn its
+PI lengths would be considerably inflated.
+At last, we evaluate the effect of different sizes of calibration and test sets under Scenario C by varying
+n and m from 100 to 200. Here four selection thresholds are included: T-test(30%), T-pos(−1.2,20%),
+T-top(60) and T-clu. The results are reported in Table 2. We see that all the three methods tend to yield
+narrowed PIs as the calibration size n increases. However, the SCOP method performs much better than
+OCP and ACP in terms of FCR control across all the settings. This clearly demonstrates the efficiency of
+our proposed SCOP, that is a data-driven method which enables FCR control with a wide range of selection
+procedures and meanwhile tends to build a relatively narrowed PIs with 1 − α level.
+4.2
+Real data applications
+4.2.1
+Drug discovery
+Early stages of drug discovery aim at finding those high binding affinity of a specific target from a pool of
+drug-target pairs (Santos et al., 2017). It is important to provide reliable PIs for those drug-target pairs
+which own high binding affinity predictions. After screening, an effective subset one may be interested in
+can be selected for further clinical trials (Huang et al., 2022). In this example, we apply the proposed SCOP
+to construct PIs of binding affinities for those promising drug-target pairs meanwhile achieve FCR control.
+We consider the DAVIS dataset (Rogers and Hahn, 2010), which contains 25, 772 drug-target pairs. Each
+pair includes the binding affinity, the structural information of drug compound and the amino acid sequence
+19
+
+Table 2: Empirical FCR (%) and AL values under Scenario C with different combinations of (n, m). The
+target FCR level is α = 10%.
+(n,m)
+T-test(30%)
+T-pos(−1.2,20%)
+T-top(60)
+T-clu
+SCOP
+OCP
+ACP
+SCOP
+OCP
+ACP
+SCOP
+OCP
+ACP
+SCOP
+OCP
+ACP
+(100,100)
+FCR
+9.70
+14.54
+4.49
+9.31
+14.48
+4.40
+9.89
+8.95
+5.41
+9.73
+15.03
+4.34
+AL
+7.22
+6.25
+8.69
+7.31
+6.25
+8.743
+6.12
+6.25
+7.31
+7.31
+6.25
+8.8
+(200,100)
+FCR
+9.89
+12.68
+3.69
+9.75
+12.48
+3.75
+10.00
+8.05
+4.62
+10.14
+13.24
+3.48
+AL
+5.75
+5.26
+7.26
+5.77
+5.26
+7.23
+4.94
+5.26
+6.14
+5.82
+5.26
+7.39
+(100,200)
+FCR
+9.60
+14.55
+4.44
+9.23
+14.63
+4.39
+9.51
+14.55
+4.44
+9.66
+15.03
+4.26
+AL
+7.32
+6.30
+8.72
+7.41
+6.30
+8.75
+7.34
+6.30
+8.72
+7.37
+6.30
+8.85
+(200,200)
+FCR
+9.88
+12.35
+3.83
+9.69
+12.18
+3.81
+9.80
+12.35
+3.83
+9.76
+12.71
+3.64
+AL
+5.76
+5.29
+7.29
+5.75
+5.29
+7.27
+5.77
+5.29
+7.29
+5.81
+5.29
+7.40
+of target protein; the drugs and targets are firstly encoded into numerical features through Python library
+DeepPurpose (Huang et al., 2020), and the responses are taken as the log-scale affinities. We randomly sample
+2,000 observations as calibration set and another 2,000 ones as test set, and use the remaining ones as training
+set to fit a small neural network model with 3 hidden layers and 5 epochs.
+Our goal is to consider building PIs of drug-target pairs on the test set through selecting their predicted
+affinities which exceed some specific threshold. Here we consider three different thresholds: 70%-quantile of
+true responses in calibration set (T-cal(70%)), 70%-quantile of predicted affinities in test set (T-test(70%)),
+and selecting those drug-target pairs with affinities larger than 9.21 while controlling FDR at 0.2 level
+(T-pos(9.21,20%)). The target FCR level is α = 10%.
+To evaluate the performance of SCOP , we also consider building PIs for those selected candidates by OCP
+and ACP. Figure 3 shows the boxplots of FCP and average length of PIs based on three methods among 100
+runs. As illustrated, the SCOP has stable FCP close to the nominal level across all the threshold selections.
+In comparison, both OCP and ACP result in conservative FCP levels. The lengths of PIs constructed by OCP
+and ACP are much broader compared to those of the SCOP. In fact, the responses of log-scale affinities truly
+range at (−5, 10), and thus those broader PIs would barely provide useful information for further clinical
+trails.
+4.2.2
+House price analysis
+We apply the SCOP to implement the prediction of house prices of interest. As an economic indicator, better
+understanding house prices can provide meaningful suggestions to researchers and decision makers in real
+estate market (Anundsen and Jansen, 2013). In recent decades, business analysts use machine learning tools
+to forecast house prices and determine the investment strategy (Park and Bae, 2015). In this example, we use
+20
+
+method
+SCOP
+OCP
+ACP
+0
+5
+10
+15
+T−cal(70%)
+T−test(70%)
+T−pos(9.21,20%)
+FCP(%)
+0
+5
+10
+15
+T−cal(70%)
+T−test(70%)
+T−pos(9.21,20%)
+Length
+Figure 3: Boxplots of the values of FCP (%) and PI length for the drug discovery example. The black dashed line
+represents the target FCR level 10% and the red rhomboid dot denotes the average value.
+our method to build PIs for those house prices which exceed certain thresholds.
+We consider one house price prediction dataset from Kaggle3, which contains 4, 251 observations after
+removing the missing data. The data records the house price and other covariates about house area, location,
+building years and so on. We randomly sample 1, 500 observations and equally split them into three parts
+as training, calibration and test sets respectively in each repetition. We firstly train a random forest model
+to predict the house prices and consider three different thresholds to select those test observations with
+high predicted house prices: T-test(70%), T-pos(0.6,20%) and T-clu. For example, the threshold T-
+pos(0.6,20%) means that one would like to select those observations with house prices larger than 0.6 million
+under the FDR control at 20% level. After selection, we construct PIs with α = 10%. Table 3 reports the
+empirical FCR level and lengths of PIs among 500 replications. We observe that both SCOP and ACP achieve
+valid FCR control, but our SCOP has more narrowed PIs compared to ACP. The FCRs of the OCP are
+much inflated which implies that many test samples with truly high house prices cannot be covered. The two
+examples demonstrate that the proposed SCOP works well for building PIs of selected samples in practical
+applications.
+3The data is available from https://www.kaggle.com/datasets/shree1992/housedata.
+21
+
+Table 3: Empirical FCRs (%) and average lengths of PIs for house price dataset with α = 10%.
+T-test(70%)
+T-pos(0.6,20%)
+T-clu
+SCOP
+OCP
+ACP
+SCOP
+OCP
+ACP
+SCOP
+OCP
+ACP
+FCR
+9.91
+21.75
+9.09
+9.78
+34.74
+7.27
+10.08
+39.71
+8.49
+AL
+1.06
+0.67
+1.12
+1.58
+0.67
+2.64
+1.72
+0.67
+1.98
+5
+Concluding remarks
+We have investigated the FCR control problem in the scenario of selective conformal inference. The validity
+of FCR-adjusted method is verified in the predictive setting, and our proposed SCOP procedure is shown to
+be widely applicable against both selection with exchangeable thresholds and non-exchangeable ranking-based
+selection with rigorous theoretical guarantee.
+To conclude, we point out several directions for future work. First, we focus mainly on using the residuals
+as nonconformity scores for the construction of PI. In fact, our framework can readily be extended to
+more general nonconformity scores such as the one based on quantile regression (Romano et al., 2019) or
+distributional regression (Chernozhukov et al., 2021). Similar theoretical results are still valid for those more
+general methods. Second, as split conformal introduces extra randomness from data splitting and reduces
+the effectiveness of training models, we may consider the implementation via Jackknife and cross-validation
+to refine the prediction intervals (Barber et al., 2021). The SCOP is applicable in such regimes, but certain
+stability conditions posed on the predictive algorithms are necessary and theoretical guarantee of SCOP
+requires further investigation. Third, it is of interest to consider adapting SCOP to the online setting, where
+one encounters an infinite sequence of samples ordered by time.
+22
+
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+26
+
+Supplementary Material for “Selective conformal inference with
+FCR control”
+A
+Auxiliary lemmas
+Variants of the following lemma often appears in the conformal inference literature (Vovk et al., 2005; Lei
+et al., 2018; Romano et al., 2019; Barber et al., 2021, 2022), which is also called the inflation of quantiles.
+Here we restate it in the deterministic form.
+Lemma A.1. Let x(⌈n(1−α)⌉) is the ⌈n(1 − α)⌉-smallest value in {xi ∈ R : i ∈ [n]}. Then for any α ∈ (0, 1),
+it holds that
+1
+n
+n
+�
+i=1
+1
+�
+xi > x(⌈n(1−α)⌉)
+�
+≤ α.
+If all values in {xi : i ∈ [n]} are distinct, it also holds that
+1
+n
+n
+�
+i=1
+1
+�
+xi > x(⌈n(1−α)⌉)
+�
+≥ α − 1
+n,
+Next lemma characterizes the change of order statistics after dropping one of the samples, which is very
+useful in the theory of conformal inference (see Lemma 2 in Romano et al. (2019)).
+Lemma A.2. For almost surely distinct random variables x1, ..., xn, let {x(r) : r ∈ [n]} be order statistics of
+{xi : i ∈ [n]}, and {x[n]\{j}
+(r)
+: r ∈ [n − 1]} be the order statistics of {xi : i ∈ [n] \ {j}}, then we have:
+(1) x(k) ≤ x[n]\{j}
+(k)
+≤ x(k+1).
+(2)
+�
+xj ≤ x(k)
+�
+= {xj ≤ x[n]\{j}
+(k)
+}.
+Lemma A.3. Suppose all values in TC ∪ TU are almost surely distinct, ˆκ(j) = ˆκ holds for any j ∈ ˆSu and
+ˆκ(j) ≤ ˆκ + Iu holds for any j ∈ U \ ˆSu, then we have
+(1) For any j ∈ U, TU\{j}
+(ˆκ(j)−Iu) ≤ TU
+(ˆκ);
+(2) For any j ∈ ˆSu, TU\{j}
+(ˆκ(j)−1) ≤ TU
+(ˆκ) ≤ TU\{j}
+(ˆκ(j)) .
+The proof of Lemma A.3 is deferred to Section D.1. The next two lemmas are corollaries of the well-known
+spacing representation of consecutive random variables (c.f. Lemma D.1), and the proofs can be found in D.2
+and D.3.
+Lemma A.4. Suppose Ui
+i.i.d.
+∼ Unif([0, 1]) and let U(1) ≤ · · · ≤ U(n) be the corresponding order statistics. For
+any absolute constant C ≥ 1, it holds that
+P
+�
+�
+max
+0≤ℓ≤n−1
+�
+U(ℓ+1) − U(ℓ)
+�
+≥
+1
+1 − 2
+�
+log(n∨m)
+n+1
+log(n ∨ m)
+n + 1
+�
+� ≤ 2(n ∨ m)−C.
+27
+
+Lemma A.5. Let ˆSc(t) = {i ∈ C : Ti ≤ t}. If
+d
+drF(R,T )(r, t) ≥ ρt holds, then for any absolute constant C ≥ 1,
+we have
+P
+�
+�
+max
+0≤ℓ≤| ˆ
+Sc(t)|−1
+�
+R
+ˆ
+Sc(t)
+(ℓ+1) − R
+ˆ
+Sc(t)
+(ℓ)
+�
+≥ 1
+ρ
+1
+1 − 2
+�
+C log(n∨m)
+| ˆ
+Sc(t)|+1
+2C log(n ∨ m)
+| ˆSc(t)| + 1
+�
+� ≤ 2(n ∨ m)−C.
+Lemma A.6 is used to bound the change in the size of the selected calibration set after changing threshold.
+We defer the proof to Section D.4. Lemma A.7 is used to lower bound the size of selected calibration set with
+arbitrary threshold t ∈ (0, 1), and Lemma A.8 is used to guarantee the threshold in SCOP will be far away
+from 0 under Assumption 3. The proofs can be found in Section D.5 and D.6.
+Lemma A.6. Let ˆSc(t2) = {i ∈ C : Ti ≤ t2} and Zi = 1 {t1 < Ti ≤ t2} − t2−t1
+t2
+for some 0 ≤ t1 < t2 ≤ 1.
+Then we have
+P
+�
+�
+1
+| ˆSc(t2)|
+������
+�
+i∈ ˆ
+Sc(t2)
+Zi
+������
+≥ 2
+�
+eC log(n ∨ m)
+| ˆSc(t2)|
+�
+t2 − t1
+t2
++ 2eC log(n ∨ m)
+| ˆSc(t2)|
+�
+� ≤ 2(n ∨ m)−C.
+Lemma A.7. Let ˆSc(t) = {i ∈ C : Ti ≤ t}. For any C ≥ 1, if 8C log(n ∨ m)/(nt) ≤ 1, we have
+P
+�
+| ˆSc(t)| ≥ n · t
+2
+�
+≤ (n ∨ m)−C.
+Lemma A.8. For any fixed γ ∈ (0, 1) and any j ∈ U, if 8C log(n ∨ m)/(mγ) ≤ 1, we have
+P
+�
+TU\{j}
+⌈γm⌉ ≤ γ
+2
+�
+≤ 2(n ∨ m)−C.
+B
+Proof of the results in Section 2
+B.1
+Proof of Proposition 2.1
+Proof. For simplicity, we assume Dt is fixed. Let Av,r be the event: r PIs are constructed, and v of these do
+not cover the corresponding true responses. Let NPIj denote the event that {Yj ̸∈ PIAD(Xj)}. It holds that
+P (Av,r) = 1
+v
+m
+�
+j=1
+P(Av,r, NPIj).
+This claim is proved by the Lemma 1 in Benjamini and Yekutieli (2005). Note that ∪r
+v=1Av,r is a disjoint
+union of events such that | ˆSu| = r and |Nu| = v, where Nu := {j ∈ �Su : Yj ̸∈ PIAD
+j
+} is the set of constructed
+PIs not covering their true labels. By the definition of FCR, we have
+FCR =
+m
+�
+r=1
+r
+�
+v=1
+v
+r P (Av,r) =
+m
+�
+r=1
+r
+�
+v=1
+1
+r
+m
+�
+j=1
+P (Av,r, NPIj) =
+m
+�
+r=1
+m
+�
+j=1
+1
+r P
+�
+| ˆSu| = r, NPIj
+�
+.
+(B.1)
+For each j ∈ ˆSu, we define the following events
+M(j)
+k
+:=
+�
+M j
+min = k
+�
+for
+k = 1, ..., m.
+28
+
+Recalling the definition of M j
+min = miny
+�
+| ˆSTj←y
+u
+| : j ∈ ˆSTj←y
+u
+�
+, we have M j
+min ≤ | ˆSu| if j ∈ ˆSu. Following
+the proof of Theorem 1 in Benjamini and Yekutieli (2005) and using the decomposition (B.1), we have
+FCR =
+m
+�
+r=1
+m
+�
+j=1
+1
+r
+m
+�
+l=1
+P
+�
+| ˆSu| = r, j ∈ ˆSu, M(j)
+l , Yj ̸∈ PIAD
+j
+(Xj)
+�
+(i)
+=
+m
+�
+r=1
+m
+�
+j=1
+r
+�
+l=1
+1
+r P
+�
+| ˆSu| = r, j ∈ ˆSu, M(j)
+l , Yi ̸∈ PIAD
+j
+(Xj)
+�
+≤
+m
+�
+j=1
+m
+�
+r=1
+r
+�
+l=1
+1
+l P
+�
+| ˆSu| = r, j ∈ ˆSu, M(j)
+l , Yi ̸∈ PIAD
+j
+(Xj)
+�
+(ii)
+=
+m
+�
+j=1
+m
+�
+l=1
+1
+l
+m
+�
+r=l
+P
+�
+| ˆSu| = r, j ∈ ˆSu, M(j)
+l , Yj ̸∈ PIAD
+j
+(Xj)
+�
+=
+m
+�
+j=1
+m
+�
+k=1
+1
+k P
+�
+M(j)
+k , Yj ̸∈ PIAD
+j
+(Xj)
+�
+(iii)
+=
+m
+�
+j=1
+m
+�
+k=1
+1
+k P
+�
+M(j)
+k
+�
+P
+�
+Yj ̸∈ PIAD
+j
+(Xj)
+�
+,
+(iv)
+≤
+m
+�
+j=1
+m
+�
+k=1
+1
+k P
+�
+M(j)
+k
+� kα
+m = α,
+where (i) holds due to Mmin(T(−j)) ≤ | ˆSu|, (ii) follows from the interchange of summations over l and r, (iii)
+holds since M (j)
+k
+is independent of the calibration set Dc and the sample j, and (iv) is true because of the
+marginal coverage guarantee that for any j ∈ [m],
+P
+�
+Yj ̸∈ PIAD
+j
+�
+≤ α∗
+j = αMmin(TU\{j})
+m
+.
+(B.2)
+Here we emphasize that the miscoverage probability P(Yj ̸∈ PIAD
+j
+(Xj)) in (iv) does not depend on the selection
+condition since the summation is over [m]. Consequently, we may utilize the exchangeability between the
+sample j and the calibration set to verify (B.2).
+B.2
+Proof of Theorem 1
+Proof. For any given non-empty subsets Su ⊆ U and Sc ⊆ C, and for any j ∈ Su, if it holds that
+α −
+1
+n + 1 ≤ P
+�
+Yj ̸∈ PIj
+��� ˆSc = Sc, ˆSu = Su
+�
+≤ α.
+(B.3)
+29
+
+Following Lemma 2.1 in Lee et al. (2016), we can decompose the FCR value with non-empty ˆSu as
+FCR0 = E
+��
+j∈ ˆ
+Su 1 {Yj ̸∈ PIj}
+| ˆSu|
+���| ˆSu| ̸= 0
+�
+= E
+�
+E
+��
+j∈ ˆ
+Su 1 {Yj ̸∈ PIj}
+| ˆSu|
+��� ˆSu, ˆSc
+� ���| ˆSu| ̸= 0
+�
+= E
+�
+� 1
+| ˆSu|
+�
+j∈ ˆ
+Su
+P
+�
+j ̸∈ PIj
+�� ˆSu, ˆSc
+� ��| ˆSu| ̸= 0
+�
+�
+≤ E
+�
+� 1
+| ˆSu|
+�
+j∈ ˆ
+Su
+α
+��| ˆSu| ̸= 0
+�
+�
+= α,
+where the last inequality holds due to the right hand side of (B.3). The FCR value can be controlled by
+FCR = FCR0 ×P
+�
+| ˆSu| > 0
+�
+≤ α.
+Similarly, using the left hand side of (B.3) we can also obtain that FCR0 ≥ α−
+1
+n+1. Further, if P(| ˆSu| > 0) = 1,
+we can also obtain the lower bound of the FCR value by
+FCR = FCR0 ×P
+�
+| ˆSu| > 0
+�
+= FCR0 ≥ α −
+1
+n + 1.
+Therefore, it suffices to verify (B.3) for SCOP under exchangeable assumption.
+According to the construction of conformal PI and Lemma A.2, given ˆSc = Sc, we know that
+{Yj ̸∈ PIj} =
+�
+Rj > QSc(1 − α)
+�
+=
+�
+Rj > QSc∪{j}(1 − α)
+�
+,
+where QSc∪{j}(1 − α) is the ⌈(1 − α)(|Sc| + 1)⌉-st smallest value in {Ri : i ∈ Sc ∪ {j}}. Next we will suppress
+the dependency on 1 − α. Invoking Lemma A.1 to Sc ∪ {j}, we have
+P
+�
+Rj > QSc∪{j}��� ˆSc = Sc, ˆSu = Su
+�
+≤ α +
+1
+|Sc| + 1E
+� �
+k∈Sc
+1
+�
+Rj > QSc∪{j}�
+− 1
+�
+Rk > QSc∪{j}� ��� ˆSu = Su, ˆSc = Sc
+�
+= α +
+1
+|Sc| + 1
+�
+k∈Sc
+∆j,k,
+(B.4)
+where
+∆j,k := P
+�
+Rj > QSc∪{j}��� ˆSc = Sc, ˆSu = Su
+�
+− P
+�
+Rk > QSc∪{j}��� ˆSc = Sc, ˆSu = Su
+�
+.
+For any j ∈ Su and k ∈ Sc, we denote
+ESu,j(ˆτ) =
+�
+�
+�
+�
+i∈Su\{j}
+{Ti ≤ ˆτ} ,
+�
+i∈U\Su
+{Ti > ˆτ}
+�
+�
+� ,
+ESc,k(ˆτ) =
+�
+�
+�
+�
+i∈Sc\{k}
+{Ti ≤ ˆτ} ,
+�
+i∈C\Sc
+{Ti > ˆτ}
+�
+�
+� .
+30
+
+Then the selection condition can be equivalently written as
+�
+ˆSc = Sc, ˆSu = Su
+�
+= {Tk ≤ ˆτ, Tj ≤ ˆτ, ESu,j(ˆτ), ESc,k(ˆτ)} .
+As a consequence, we have
+∆j,k = P
+�
+Rj > QSc∪{j}|Tk ≤ ˆτ, Tj ≤ ˆτ, ESu,j(ˆτ), ESc,k(ˆτ)
+�
+− P
+�
+Rk > QSc∪{j}|Tk ≤ ˆτ, Tj ≤ ˆτ, ESu,j(ˆτ), ESc,k(ˆτ)
+�
+.
+(B.5)
+Let ˆτ(j,k) be the threshold obtained by swapping sample j ∈ U and k ∈ C. Under our assumption, we know
+ˆτ(j,k) = ˆτ with probability 1. It follows that
+{Tk ≤ ˆτ, Tj ≤ ˆτ, ESu,j(ˆτ), ESc,k(ˆτ)} =
+�
+Tk ≤ ˆτ(j,k), Tj ≤ ˆτ(j,k), ESu,j(ˆτ(j,k)), ESc,k(ˆτ(j,k))
+�
+.
+Therefore, we can guarantee ∆j,k = 0 from (B.5).
+C
+Proof of the results in Section 3
+In this section, we use Eu,−j[·] and Pu,−j(·) to denote the expectation and probability given data set Du,−j.
+C.1
+Proof of Lemma 1
+Proof. According to the definition of FCR, we have
+FCR = E
+�
+� 1
+| ˆSu|
+�
+j∈ ˆ
+Su
+1 {Yj ̸∈ PIj}
+�
+� (i)
+= E
+�
+��
+j∈U
+1
+�
+j ∈ ˆSu
+�
+ˆκ
+1 {Yj ̸∈ PIj}
+�
+�
+(ii)
+= E
+�
+��
+j∈U
+1
+ˆκ(j) 1
+�
+j ∈ ˆSu
+�
+1 {Yj ̸∈ PIj}
+�
+�
+= E
+�
+��
+j∈U
+1
+ˆκ(j) Pu,−j
+�
+Yj ̸∈ PIj
+��j ∈ ˆSu
+�
+Pu,−j
+�
+j ∈ ˆSu
+�
+�
+�
+≤ E
+�
+��
+j∈U
+(α + ∆(Du,−j))
+1
+�
+j ∈ ˆSu
+�
+ˆκ(j)
+�
+�
+(iii)
+= α · E
+�
+��
+j∈U
+1
+�
+j ∈ ˆSu
+�
+| ˆSu|
+�
+� + E
+�
+� 1
+| ˆSu|
+�
+j∈ ˆ
+Su
+∆(Du,−j)
+�
+�
+= α + E
+�
+� 1
+| ˆSu|
+�
+j∈ ˆ
+Su
+∆(Du,−j)
+�
+� ,
+where the equality (i) holds due to | ˆSu| = ˆκ (c.f. the definition of ˆSu in (8)), (ii) and (iii) come from the
+assumption ˆκ = ˆκ(j) under the event j ∈ ˆSu. The lower bound can be derived similarly.
+31
+
+C.2
+Self-driven selection
+In this subsection, we provide the proofs for the results of self-driven selection procedures, where the ranking
+threshold ˆκ only depends on the test set Du.
+C.2.1
+Proof of Theorem 2
+Proof of Theorem 2. Recall the definitions
+ˆS(j)
+c
+=
+�
+i ∈ C : Ti ≤ TU\{j}
+(ˆκ(j))
+�
+,
+Q
+ˆ
+S(j)
+c
+∪{j} = R
+ˆ
+S(j)
+c
+∪{j}
+(⌈(1−α)(| ˆ
+S(j)
+c
+|+1)⌉).
+Invoking Lemma A.1, we know it holds that
+α −
+1
+| ˆS(j)
+c | + 1
+≤
+1
+| ˆS(j)
+c | + 1
+�
+i∈ ˆ
+S(j)
+c
+∪{j}
+1
+�
+Ri > Q
+ˆ
+S(j)
+c
+∪{j}�
+≤ α.
+(C.1)
+In addition, we also know {j ̸∈ PIj} = {Rj > R
+ˆ
+S(j)
+c
+(⌈(1−α)(| ˆ
+S(j)
+c
+|+1)⌉)} = {Rj > R
+ˆ
+S(j)
+c
+∪{j}
+(⌈(1−α)(| ˆ
+S(j)
+c
+|+1)⌉)} by the
+construction of PIj and Lemma A.2. Rearranging the inequality in the right hand side of (C.1) gives
+1 {j ̸∈ PIj} = 1
+�
+Rj > Q
+ˆ
+Sc∪{j}�
+≤ α −
+1
+| ˆS(j)
+c | + 1
+�
+i∈ ˆ
+S(j)
+c
+∪{j}
+1
+�
+Ri > Q
+ˆ
+S(j)
+c
+∪{j}�
++ 1
+�
+Rj > Q
+ˆ
+Sc∪{j}�
+= α −
+1
+| ˆS(j)
+c | + 1
+�
+k∈ ˆ
+S(j)
+c
+�
+1
+�
+Rk > Q
+ˆ
+S(j)
+c
+∪{j}�
+− 1
+�
+Rj > Q
+ˆ
+S(j)
+c
+∪{j}��
++ 1
+�
+Rj > Q
+ˆ
+Sc∪{j}�
+− 1
+�
+Rj > Q
+ˆ
+S(j)
+c
+∪{j}�
+.
+(C.2)
+Given the dataset Du,−j, taking expectation on both sides of (C.2) conditional on the event {j ∈ ˆSu} (that is
+{Tj ≤ TU
+(ˆκ)}) yields
+Pu,−j
+�
+Yj ̸∈ PIj
+���Tj ≤ TU
+(ˆκ)
+�
+≤ α − Eu,−j
+�
+�
+1
+| ˆS(j)
+c | + 1
+�
+k∈ ˆ
+S(j)
+c
+�
+1
+�
+Rk > Q
+ˆ
+S(j)
+c
+∪{j}�
+− 1
+�
+Rj > Q
+ˆ
+S(j)
+c
+∪{j}�� ���Tj ≤ TU
+(ˆκ)
+�
+�
++ Pu,−j
+�
+Rj > Q
+ˆ
+Sc∪{j}���Tj ≤ TU
+(ˆκ)
+�
+− Pu,−j
+�
+Rj > Q
+ˆ
+S(j)
+c
+∪{j}���Tj ≤ TU
+(ˆκ)
+�
+.
+(C.3)
+Rearranging the inequality in the left hand side of (C.1), we can also have
+Pu,−j
+�
+Yj ̸∈ PIj
+���Tj ≤ TU
+(ˆκ)
+��
+≥ α −
+1
+| ˆS(j)
+c | + 1
+− Eu,−j
+�
+�
+1
+| ˆS(j)
+c | + 1
+�
+k∈ ˆ
+S(j)
+c
+�
+1
+�
+Rk > Q
+ˆ
+S(j)
+c
+∪{j}�
+− 1
+�
+Rj > Q
+ˆ
+S(j)
+c
+∪{j}�� ���Tj ≤ TU
+(ˆκ)
+�
+�
++ Pu,−j
+�
+Rj > Q
+ˆ
+Sc∪{j}���Tj ≤ TU
+(ˆκ)
+�
+− Pu,−j
+�
+Rj > Q
+ˆ
+S(j)
+c
+∪{j}���Tj ≤ TU
+(ˆκ)
+�
+.
+(C.4)
+Next we introduce three lemmas to control the additional terms except α in (C.3) and (C.4). And we
+defer their proofs to Section C.3.1, C.3.2 and C.3.3.
+32
+
+Lemma C.1. Under the conditions of Theorem 2, we have
+������
+Eu,−j
+�
+�
+1
+| ˆS(j)
+c | + 1
+�
+k∈ ˆ
+S(j)
+c
+�
+1
+�
+Rj > Q
+ˆ
+S(j)
+c
+∪{j}�
+− 1
+�
+Rk > Q
+ˆ
+S(j)
+c
+∪{j}�� ���Tj ≤ TU
+(ˆκ)
+�
+�
+������
+≤ 2
+TU\{j}
+(ˆκ(j)) − TU\{j}
+(ˆκ(j)−Iu)
+TU\{j}
+(ˆκ(j)−Iu)
+.
+Lemma C.2. Under the conditions of Theorem 2, we have
+���Pu,−j
+�
+Rj > Q
+ˆ
+Sc∪{j}���Tj ≤ TU
+(ˆκ)
+�
+− Pu,−j
+�
+Rj > Q
+ˆ
+S(j)
+c
+∪{j}���Tj ≤ TU
+(ˆκ)
+����
+≤ 4C log(n ∨ m)
+ρTU\{j}
+(ˆκ(j)−Iu)
+�
+�12C log(n ∨ m)
+nTU\{j}
+(ˆκ(j))
++
+2
+�
+TU\{j}
+(ˆκ(j)) − TU\{j}
+(ˆκ(j)−1)
+�
+TU\{j}
+(ˆκ(j))
+�
+� + 3(n ∨ m)−C
+TU\{j}
+(ˆκ(j)−Iu)
+.
+Lemma C.3. Under the conditions of Theorem 2, we have
+Eu,−j
+�
+1
+| ˆS(j)
+c | + 1
+���Tj ≤ TU
+(ˆκ)
+�
+≤
+TU\{j}
+(ˆκ(j))
+TU\{j}
+(ˆκ(j)−Iu)
+�
+�
+2
+nTU\{j}
+(ˆκ(j)) + 2
++ (n ∨ m)−C
+�
+� .
+Armed with Lemmas C.1, C.2 and C.3, we can obtain the result of Theorem 2 after some simplifications.
+A remark on the construction of virtual calibration set.
+To decouple the dependence on the candidate
+j, we introduce the virtual calibration set ˆS(j)
+c
+by using the threshold TU\{j}
+(ˆκ(j)) , where ˆκ(j) is independent of
+test sample j. Another good property of ˆS(j)
+c
+is that ˆSc ⊆ ˆS(j)
+c . In Assumption 3, we assume ˆκ ≥ γm holds
+almost surely. Let us consider the following naive virtual calibration set,
+ˆSnaive
+c
+:=
+�
+i ∈ C : Ti ≤ TU\{j}
+(⌈γm⌉)
+�
+.
+Even though ˆSnaive
+c
+possesses two good properties of ˆS(j)
+c , but we cannot use ˆSnaive
+c
+in our proof. Notice that
+| ˆSnaive
+c
+| − | ˆSc| =
+�
+i∈ ˆ
+Snaive
+c
+1
+�
+TU
+(ˆκ) < Ti ≤ TU\{j}
+(⌈γm⌉)
+�
+.
+Given Du,−j, the conditional expectation of the indicator function is
+Eu,−j
+�
+1
+�
+TU
+(ˆκ) < Ti ≤ TU\{j}
+(⌈γm⌉)
+�
+|Ti ≤ TU\{j}
+(⌈γm⌉)
+�
+=
+TU\{j}
+(⌈γm⌉) − TU
+(ˆκ)
+TU\{j}
+(⌈γm⌉)
+.
+Since the ranking threshold ˆκ can be very close to m, the conditional expectation above may scale as Op(1)
+according to the spacing representation (c.f. Lemma D.1). As a consequence, we cannot have an op(1) gap for
+FCR control. In Section C.3.2, we show the corresponding conditional expectation of our careful chosen ˆS(j)
+c
+scales as Op( log(n∨m)
+m
+). This explains why we cannot use ˆSnaive
+c
+as the virtual calibration set in the proof, and
+ˆS(j)
+c
+is indeed a nontrivial construction.
+33
+
+C.2.2
+Proof of Theorem 3
+Proof. Recall that,
+∆(Du,−j) = 8C log(n ∨ m)
+ρTU\{j}
+(ˆκ(j)−Iu)
+�
+�12C log(n ∨ m)
+nTU\{j}
+(ˆκ(j))
++
+2
+�
+TU\{j}
+(ˆκ(j)) − TU\{j}
+(ˆκ(j)−1)
+�
+TU\{j}
+(ˆκ(j))
+�
+� + 2
+TU\{j}
+(ˆκ(j)) − TU\{j}
+(ˆκ(j)−Iu)
+TU\{j}
+(ˆκ(j)−Iu)
+.
+From Lemma A.4, we can guarantee that
+TU\{j}
+(ˆκ(j)) − TU\{j}
+(ˆκ(j)−Iu) ≤ Iu ×
+max
+1≤ℓ≤m−2
+�
+TU\{j}
+(ℓ+1) − TU\{j}
+(ℓ)
+�
+≤ 2CIu log(n ∨ m)
+m
+,
+holds with probability at least 1 − (n ∨ m)−C. In addition, by Lemma A.8, we have
+TU\{j}
+(ˆκ(j)−Iu) ≥ TU\{j}
+(⌈γm⌉−Iu) ≥ TU
+(⌈γm⌉−Iu) ≥ TU
+(⌈(γ−Iu/m)m⌉) ≥ γ − Iu/m
+2
+holds with probability at least 1 − (n ∨ m)−C. Then we have
+E
+�
+max
+1≤j≤m ∆(Du,−j)
+�
+≲
+log2(n ∨ m)
+ργ(γ − Iu/m)
+� 1
+m + 1
+n
+�
++ Iu log(n ∨ m)
+m(γ − Iu/m).
+Then the conclusion follows from Lemma 1 immediately.
+C.3
+Deferred Proofs of Section C.2
+C.3.1
+Proof of Lemma C.1
+Proof. We first notice that
+Eu,−j
+�
+�
+1
+| ˆS(j)
+c | + 1
+�
+k∈ ˆ
+S(j)
+c
+�
+1
+�
+Rj > Q
+ˆ
+S(j)
+c
+∪{j}�
+− 1
+�
+Rk > Q
+ˆ
+S(j)
+c
+∪{j}�� ���j ∈ ˆSu
+�
+�
+=
+�
+Sc⊆[m]
+Pu,−j
+�
+ˆS(j)
+c
+= Sc
+�
+|Sc| + 1
+×
+Eu,−j
+� �
+k∈Sc
+1
+�
+Rj > QSc∪{j}�
+− 1
+�
+Rk > QSc∪{j}� ���Tj ≤ TU
+(ˆκ), ˆS(j)
+c
+= Sc
+�
+.
+(C.5)
+Next we will bound the conditional expectation term in (C.5). From the definition of ˆS(j)
+c , we know
+{ ˆS(j)
+c
+= Sc} =
+� �
+k∈Sc
+{Tk ≤ TU\{j}
+(ˆκ(j)) }
+� �
+�
+�
+�
+k∈C\Sc
+{Tk > TU\{j}
+(ˆκ(j)) }
+�
+� .
+Since both sample j ∈ ˆSu and k ∈ Sc are independent of TU\{j}
+(ˆκ(j)) , we can guarantee that
+Pu,−j
+�
+Rk > Q
+ˆ
+S(j)
+c
+∪{j}�� ˆS(j)
+c
+= Sc, Tj ≤ TU\{j}
+(ˆκ(j))
+�
+=
+Pu,−j
+�
+Rk > QSc∪{j}, �
+i∈Sc∪{j}
+�
+Ti ≤ TU\{j}
+(ˆκ(j))
+��
+Pu,−j
+��
+i∈Sc∪{j}
+�
+Ti ≤ TU\{j}
+(ˆκ(j))
+��
+=
+Pu,−j
+�
+Rj > QSc∪{j}, �
+i∈Sc∪{j}
+�
+Ti ≤ TU\{j}
+(ˆκ(j))
+��
+Pu,−j
+��
+i∈Sc∪{j}
+�
+Ti ≤ TU\{j}
+(ˆκ(j))
+��
+= Pu,−j
+�
+Rj > Q
+ˆ
+S(j)
+c
+∪{j}�� ˆS(j)
+c
+= Sc, Tj ≤ TU\{j}
+(ˆκ(j))
+�
+,
+(C.6)
+34
+
+where the penultimate equality holds since the distribution of (Xj, Yj) and (Xk, Yk) are same. It follows that
+Eu,−j
+� �
+k∈Sc
+1
+�
+Rj > Q
+ˆ
+S(j)
+c
+∪{j}�
+− 1
+�
+Rk > QSc∪{j}� ���Tj ≤ TU
+(ˆκ), ˆS(j)
+c
+= Sc
+�
+=
+�
+k∈Sc
+�
+Pu,−j
+�
+Rk > QSc∪{j}���Tj ≤ TU\{j}
+(ˆκ(j)) , ˆS(j)
+c
+= Sc
+�
+− Pu,−j
+�
+Rk > QSc∪{j}���Tj ≤ TU
+(ˆκ), ˆS(j)
+c
+= Sc
+� �
+−
+|Sc|
+|Sc| + 1
+�
+Pu,−j
+�
+Rj > QSc∪{j}���Tj ≤ TU\{j}
+(ˆκ(j)) , ˆS(j)
+c
+= Sc
+�
+− Pu,−j
+�
+Rj > QSc∪{j}���Tj ≤ TU
+(ˆκ), ˆS(j)
+c
+= Sc
+� �
++
+�
+k∈Sc
+�
+Pu,−j
+�
+Rk > QSc∪{j}���Tj ≤ TU\{j}
+(ˆκ(j)) , ˆS(j)
+c
+= Sc
+�
+− Pu,−j
+�
+Rj > QSc∪{j}���Tj ≤ TU\{j}
+(ˆκ(j)) , ˆS(j)
+c
+= Sc
+� �
+=:
+�
+k∈Sc
+∆k +
+|Sc|
+|Sc| + 1∆j + 0.
+(C.7)
+For ∆k, we have the following upper bound
+∆k =
+Pu,−j
+�
+Rk > QSc∪{j}, Tj ≤ TU\{j}
+(ˆκ(j)) , ˆS(j)
+c
+= Sc
+�
+Pu,−j
+�
+Tj ≤ TU\{j}
+(ˆκ(j)) , ˆS(j)
+c
+= Sc
+�
+−
+Pu,−j
+�
+Rk > QSc∪{j}, Tj ≤ TU
+(ˆκ), ˆS(j)
+c
+= Sc
+�
+Pu,−j
+�
+Tj ≤ TU
+(ˆκ), ˆS(j)
+c
+= Sc
+�
+(i)
+≤
+Pu,−j
+�
+Rk > QSc∪{j}, TU\{j}
+(ˆκ(j)−Iu) < Tj ≤ TU\{j}
+(ˆκ(j)) , ˆS(j)
+c
+= Sc
+�
+Pu,−j
+�
+Tj ≤ TU
+(ˆκ), ˆS(j)
+c
+= Sc
+�
+(ii)
+≤
+Pu,−j
+�
+TU\{j}
+(ˆκ(j)−Iu) < Tj ≤ TU\{j}
+(ˆκ(j)) , ˆS(j)
+c
+= Sc
+�
+Pu,−j
+�
+Tj ≤ TU\{j}
+(ˆκ)
+, ˆS(j)
+c
+= Sc
+�
+(iii)
+≤
+Pu,−j
+�
+TU\{j}
+(ˆκ(j)−Iu) < Tj ≤ TU\{j}
+(ˆκ(j)) , ˆS(j)
+c
+= Sc
+�
+Pu,−j
+�
+Tj ≤ TU\{j}
+(ˆκ(j)−Iu), ˆS(j)
+c
+= Sc
+�
+(iv)
+=
+TU\{j}
+(ˆκ(j)) − TU\{j}
+(ˆκ(j)−Iu)
+TU\{j}
+(ˆκ(j)−Iu)
+,
+(C.8)
+where (i) and (iii) hold since TU\{j}
+(ˆκ(j)−Iu) ≤ TU
+(ˆκ) ≤ TU\{j}
+(ˆκ(j)) (c.f. Lemma A.3), (ii) follows from the conclusion
+(2) of Lemma A.2, and (iv) follows from the with firstly conditioning on Du,−j ∪ Dc. Similarly, we can obtain
+35
+
+the following lower bound
+∆k =
+Pu,−j
+�
+Rk > QSc∪{j}, Tj ≤ TU\{j}
+(ˆκ(j)) , ˆS(j)
+c
+= Sc
+�
+Pu,−j
+�
+Tj ≤ TU\{j}
+(ˆκ(j)) , ˆS(j)
+c
+= Sc
+�
+−
+Pu,−j
+�
+Rk > QSc∪{j}, Tj ≤ TU
+(ˆκ), ˆS(j)
+c
+= Sc
+�
+Pu,−j
+�
+Tj ≤ TU
+(ˆκ), ˆS(j)
+c
+= Sc
+�
+(v)
+≥
+Pu,−j
+�
+Rk > QSc∪{j}, Tj ≤ TU
+(ˆκ), ˆS(j)
+c
+= Sc
+�
+Pu,−j
+�
+Tj ≤ TU
+(ˆκ), ˆS(j)
+c
+= Sc
+�
+�
+�
+Pu,−j
+�
+Tj ≤ TU
+(ˆκ), ˆS(j)
+c
+= Sc
+�
+Pu,−j
+�
+Tj ≤ TU\{j}
+(ˆκ(j)) , ˆS(j)
+c
+= Sc
+� − 1
+�
+�
+(vi)
+≥ −
+Pu,−j
+�
+TU
+(ˆκ) < Tj ≤ TU\{j}
+(ˆκ(j)) , ˆS(j)
+c
+= Sc
+�
+Pu,−j
+�
+Tj ≤ TU\{j}
+(ˆκ(j)) , ˆS(j)
+c
+= Sc
+�
+≥ −
+TU\{j}
+(ˆκ(j)) − TU\{j}
+(ˆκ(j)−Iu)
+TU\{j}
+(ˆκ(j))
+,
+(C.9)
+where (v) holds because TU
+(ˆκ) ≤ TU\{j}
+(ˆκ(j)) , and (vi) is true since the term in the bracket is negative. Combining
+(C.8) and (C.9), we have
+|∆k| ≤
+TU\{j}
+(ˆκ(j)) − TU\{j}
+(ˆκ(j)−Iu)
+TU\{j}
+(ˆκ(j)−Iu)
+.
+(C.10)
+For ∆j, we have the following upper bound
+∆j =
+Pu,−j
+�
+Rj > QSc∪{j}, Tj ≤ TU
+(ˆκ), ˆS(j)
+c
+= Sc
+�
+Pu,−j
+�
+Tj ≤ TU
+(ˆκ), ˆS(j)
+c
+= Sc
+�
+−
+Pu,−j
+�
+Rj > QSc∪{j}, Tj ≤ TU\{j}
+(ˆκ(j)) , ˆS(j)
+c
+= Sc
+�
+Pu,−j
+�
+Tj ≤ TU\{j}
+(ˆκ(j)) , ˆS(j)
+c
+= Sc
+�
+≤
+Pu,−j
+�
+Rj > QSc∪{j}, Tj ≤ TU\{j}
+(ˆκ(j)) , ˆS(j)
+c
+= Sc
+�
+Pu,−j
+�
+Tj ≤ TU\{j}
+(ˆκ(j)) , ˆS(j)
+c
+= Sc
+�
+�
+�
+Pu,−j
+�
+Tj ≤ TU\{j}
+(ˆκ(j)) , ˆS(j)
+c
+= Sc
+�
+Pu,−j
+�
+Tj ≤ TU
+(ˆκ), ˆS(j)
+c
+= Sc
+�
+− 1
+�
+�
+≤
+Pu,−j
+�
+Tj ≤ TU\{j}
+(ˆκ(j)) , ˆS(j)
+c
+= Sc
+�
+Pu,−j
+�
+Tj ≤ TU
+(ˆκ), ˆS(j)
+c
+= Sc
+�
+− 1
+≤
+TU\{j}
+(ˆκ(j)) − TU\{j}
+(ˆκ(j)−Iu)
+TU\{j}
+(ˆκ(j)−Iu)
+.
+The lower bound of ∆j can be derived as
+∆j ≥
+Pu,−j
+�
+Rj > QSc∪{j}, Tj ≤ TU
+(ˆκ), ˆS(j)
+c
+= Sc
+�
+Pu,−j
+�
+Tj ≤ TU\{j}
+(ˆκ(j)) , ˆS(j)
+c
+= Sc
+�
+−
+Pu,−j
+�
+Rj > QSc∪{j}, Tj ≤ TU\{j}
+(ˆκ(j)) , ˆS(j)
+c
+= Sc
+�
+Pu,−j
+�
+Tj ≤ TU\{j}
+(ˆκ(j)) , ˆS(j)
+c
+= Sc
+�
+≥ −
+Pu,−j
+�
+TU\{j}
+(ˆκ(j)−Iu) < Tj ≤ TU\{j}
+(ˆκ(j)) , ˆS(j)
+c
+= Sc
+�
+Pu,−j
+�
+Tj ≤ TU\{j}
+(ˆκ(j)) , ˆS(j)
+c
+= Sc
+�
+≥ −
+TU\{j}
+(ˆκ(j)) − TU\{j}
+(ˆκ(j)−Iu)
+TU\{j}
+(ˆκ(j))
+.
+36
+
+Hence we have
+|∆j| ≤
+TU\{j}
+(ˆκ(j)) − TU\{j}
+(ˆκ(j)−Iu)
+TU\{j}
+(ˆκ(j)−Iu)
+.
+(C.11)
+Plugging (C.10) and (C.11) into (C.7), we have
+�����Eu,−j
+� �
+k∈Sc
+1
+�
+Rj > Q
+ˆ
+S(j)
+c
+∪{j}�
+− 1
+�
+Rk > QSc∪{j}� ���Tj ≤ TU
+(ˆκ), ˆS(j)
+c
+= Sc
+������
+≤ (|Sc| + 1) ·
+TU\{j}
+(ˆκ(j)) − TU\{j}
+(ˆκ(j)−Iu)
+TU\{j}
+(ˆκ(j)−Iu)
+.
+Taking consideration of (C.5), we can complete the proof of Lemma C.1.
+C.3.2
+Proof of Lemma C.2
+Proof. From TU
+(ˆκ) ≤ TU\{j}
+(ˆκ(j)) in Lemma A.3, we know ˆSc ⊆ ˆS(j)
+c . Then for any j ∈ ˆSu, we have
+| ˆS(j)
+c | − | ˆSc| =
+�
+i∈ ˆ
+S(j)
+c
+1
+�
+TU
+(ˆκ) < Ti ≤ TU\{j}
+(ˆκ(j))
+�
+≤
+�
+i∈ ˆ
+S(j)
+c
+1
+�
+TU\{j}
+(ˆκ(j)−1) < Ti ≤ TU\{j}
+(ˆκ(j))
+�
+=: d(j),
+(C.12)
+where the inequality holds since TU\{j}
+(ˆκ(j)−1) ≤ TU
+(ˆκ) holds for j ∈ ˆSu (c.f. Lemma A.3). Recall that
+Q
+ˆ
+Sc∪{j} = R
+ˆ
+Sc∪{j}
+(⌈(1−α)(| ˆ
+Sc|+1)⌉),
+Q
+ˆ
+S(j)
+c
+∪{j} = R
+ˆ
+S(j)
+c
+∪{j}
+(⌈(1−α)(| ˆ
+S(j)
+c
+|+1)⌉).
+Next we bound the rank of value R
+ˆ
+Sc
+(⌈(1−α)(| ˆ
+Sc|+1)⌉) in the set ˆS(j)
+c . It follows from ˆSc ⊆ ˆS(j)
+c
+and (C.12) that
+�
+i∈ ˆ
+S(j)
+c
+1
+�
+Ri ≤ R
+ˆ
+Sc
+(⌈(1−α)(| ˆ
+Sc|+1)⌉)
+�
+≥
+�
+i∈ ˆ
+Sc
+1
+�
+Ri ≤ R
+ˆ
+Sc
+(⌈(1−α)(| ˆ
+Sc|+1)⌉)
+�
+= ⌈(1 − α)(| ˆSc| + 1)⌉
+≥ ⌈(1 − α)(| ˆS(j)
+c | − d(j) + 1)⌉,
+and
+�
+i∈ ˆ
+S(j)
+c
+1
+�
+Ri ≤ R
+ˆ
+Sc
+(⌈(1−α)(| ˆ
+Sc|+1)⌉)
+�
+≤ d(j) +
+�
+i∈ ˆ
+Sc
+1
+�
+Ri ≤ R
+ˆ
+Sc
+(⌈(1−α)(| ˆ
+Sc|+1)⌉)
+�
+= d(j) + ⌈(1 − α)(| ˆSc| + 1)⌉
+≤ d(j) + ⌈(1 − α)(| ˆS(j)
+c | + 1)⌉.
+Two bounds above indicate that
+R
+ˆ
+S(j)
+c
+(⌈(1−α)(| ˆ
+S(j)
+c
+|−d(j)+1)) ≤ R
+ˆ
+Sc
+(⌈(1−α)(| ˆ
+Sc|+1)⌉) ≤ R
+ˆ
+S(j)
+c
+(d(j)+⌈(1−α)(| ˆ
+S(j)
+c
+|+1)).
+(C.13)
+37
+
+By the right hand side of (C.13), we have
+Pu,−j
+�
+Rj > Q
+ˆ
+Sc∪{j}���Tj ≤ TU
+(ˆκ)
+�
+− Pu,−j
+�
+Rj > Q
+ˆ
+S(j)
+c
+∪{j}���Tj ≤ TU
+(ˆκ)
+�
+(i)
+= Pu,−j
+�
+Rj > Q
+ˆ
+Sc
+���Tj ≤ TU
+(ˆκ)
+�
+− Pu,−j
+�
+Rj > Q
+ˆ
+S(j)
+c
+���Tj ≤ TU
+(ˆκ)
+�
+≥ −Eu,−j
+�
+1
+�
+Rj > R
+ˆ
+S(j)
+c
+(⌈(1−α)(| ˆ
+S(j)
+c
+|+1)⌉)
+�
+− 1
+�
+Rj > R
+ˆ
+S(j)
+c
+(⌈(1−α)(| ˆ
+S(j)
+c
+|+1)⌉+d(j))
+� ���Tj ≤ TU
+(ˆκ)
+�
+(ii)
+≥ −
+Pu,−j
+�
+R
+ˆ
+S(j)
+c
+(⌈(1−α)(| ˆ
+S(j)
+c
+|+1)⌉) < Rj ≤ R
+ˆ
+S(j)
+c
+(⌈(1−α)(| ˆ
+S(j)
+c
+|+1)⌉+d(j)), Tj ≤ TU\{j}
+(ˆκ(j))
+�
+Pu,−j
+�
+Tj ≤ TU\{j}
+(ˆκ(j)−Iu)
+�
+(iii)
+≥ −
+Eu,−j
+�
+R
+ˆ
+S(j)
+c
+(⌈(1−α)(| ˆ
+S(j)
+c
+|+1)⌉+d(j)) − R
+ˆ
+S(j)
+c
+(⌈(1−α)(| ˆ
+S(j)
+c
+|+1)⌉)
+�
+TU\{j}
+(ˆκ(j)−Iu)
+,
+(C.14)
+where (i) comes from the conclusion (2) of Lemma A.2 by dropping j, (ii) holds due to TU\{j}
+(ˆκ(j)−Iu) ≤ TU
+(ˆκ) ≤
+TU\{j}
+(ˆκ(j)) for any j ∈ U (c.f. Lemma A.3), and (iii) holds by dropping event {Tj ≤ TU\{j}
+(ˆκ(j)) }. By the left hand
+side of (C.13), we similarly have
+Pu,−j
+�
+Rj > Q
+ˆ
+Sc∪{j}���Tj ≤ TU
+(ˆκ)
+�
+− Pu,−j
+�
+Rj > Q
+ˆ
+S(j)
+c
+∪{j}���Tj ≤ TU
+(ˆκ)
+�
+≤
+Eu,−j
+�
+R
+ˆ
+S(j)
+c
+(⌈(1−α)(| ˆ
+S(j)
+c
+|+1)⌉) − R
+ˆ
+S(j)
+c
+(⌈(1−α)(| ˆ
+S(j)
+c
+|+1−d(j))⌉)
+�
+TU\{j}
+(ˆκ(j)−Iu)
+.
+(C.15)
+Applying Lemma A.7, with probability at least 1 − (n ∨ m)−C, it holds
+| ˆS(j)
+c | ≥ n · TU\{j}
+(ˆκ(j)) − C
+�
+n log(n ∨ m)
+�
+TU\{j}
+(ˆκ(j))
+�
+1 − TU\{j}
+(ˆκ(j))
+�
+≥ n · TU\{j}
+(ˆκ(j))
+�
+�1 − C
+�
+�
+�
+�log(n ∨ m)
+nTU\{j}
+(ˆκ(j))
+�
+� ≥ 1
+2n · TU\{j}
+(ˆκ(j)) ,
+(C.16)
+where we used the assumption 8C log(n ∨ m)/(nTU\{j}
+(ˆκ(j)) ) ≤ 1 almost surely. Given Du,−j, applying Lemma
+A.6 with t1 = TU\{j}
+(ˆκ(j)−1) and t2 = TU\{j}
+(ˆκ(j)) , we can guarantee that
+d(j)
+| ˆS(j)
+c |
+=
+1
+| ˆS(j)
+c |
+�
+i∈ ˆ
+S(j)
+c
+�
+�1
+�
+TU\{j}
+(ˆκ(j)−1) < Ti ≤ TU\{j}
+(ˆκ(j))
+�
+−
+TU\{j}
+(ˆκ(j)) − TU\{j}
+(ˆκ(j)−1)
+TU\{j}
+(ˆκ(j))
+�
+� +
+TU\{j}
+(ˆκ(j)) − TU\{j}
+(ˆκ(j)−1)
+TU\{j}
+(ˆκ(j))
+≤ 2
+�
+eC log(n ∨ m)
+| ˆS(j)
+c |
+�
+�
+�
+�
+�
+TU\{j}
+(ˆκ(j)) − TU\{j}
+(ˆκ(j)−1)
+TU\{j}
+(ˆκ(j))
++ 2eC log(n ∨ m)
+| ˆS(j)
+c |
++
+TU\{j}
+(ˆκ(j)) − TU\{j}
+(ˆκ(j)−1)
+TU\{j}
+(ˆκ(j))
+≤ 6
+�
+�
+�
+�C log(n ∨ m)
+nTU\{j}
+(ˆκ(j))
+�
+�
+�
+�
+�
+TU\{j}
+(ˆκ(j)) − TU\{j}
+(ˆκ(j)−1)
+TU\{j}
+(ˆκ(j))
++ 6C log(n ∨ m)
+nTU\{j}
+(ˆκ(j))
++
+TU\{j}
+(ˆκ(j)) − TU\{j}
+(ˆκ(j)−1)
+TU\{j}
+(ˆκ(j))
+≤ 12C log(n ∨ m)
+nTU\{j}
+(ˆκ(j))
++
+2
+�
+TU\{j}
+(ˆκ(j)) − TU\{j}
+(ˆκ(j)−1)
+�
+TU\{j}
+(ˆκ(j))
+=: ∆1(Du,−j),
+(C.17)
+38
+
+holds with probability at least 1 − 2(n ∨ m)−C. Given Du,−j, with probability at least 1 − 3(n ∨ m)−C, we
+have
+R
+ˆ
+S(j)
+c
+(⌈(1��α)(| ˆ
+S(j)
+c
+|+1)⌉) − R
+ˆ
+S(j)
+c
+(⌈(1−α)(| ˆ
+S(j)
+c
+|−d(j)+1)⌉) =
+⌈(1−α)(| ˆ
+S(j)
+c
+|+1)⌉+d(j)−1
+�
+ℓ=⌈(1−α)(| ˆ
+S(j)
+c
+|−d(j)+1)⌉
+R
+ˆ
+S(j)
+c
+(ℓ+1) − R
+ˆ
+S(j)
+c
+(ℓ)
+≤ d(j)
+max
+0≤ℓ≤| ˆ
+S(j)
+c
+|
+�
+R
+ˆ
+S(j)
+c
+(ℓ+1) − R
+ˆ
+S(j)
+c
+(ℓ)
+�
+(i)
+≤
+d(j)
+| ˆS(j)
+c |
+· 1
+ρ
+| ˆS(j)
+c |
+1 − 2
+�
+C log(n∨m)
+| ˆ
+S(j)
+c
+|+1
+2C log(n ∨ m)
+| ˆSc(t)| + 1
+(ii)
+≤ ∆1(Du,−j) · 1
+ρ
+2C log(n ∨ m)
+1 − 2
+�
+C log(n∨m)
+| ˆ
+S(j)
+c
+|+1
+(iii)
+≤ 4C log(n ∨ m)
+ρ
+∆1(Du,−j),
+(C.18)
+where (i) follows from applying Lemma A.5 with t = TU\{j}
+(ˆκ(j)) , (ii) comes from (C.17), and (iii) comes from
+(C.16) and the assumption 8C log(n ∨ m)/(nTU\{j}
+(ˆκ(j)) ) ≤ 1 almost surely. Similarly, we also have
+R
+ˆ
+S(j)
+c
+(⌈(1−α)(| ˆ
+S(j)
+c
+|+1)⌉+d(j)) − R
+ˆ
+S(j)
+c
+(⌈(1−α)(| ˆ
+S(j)
+c
+|+1)⌉) ≤ d(j)
+max
+0≤ℓ≤| ˆ
+S(j)
+c
+|
+�
+R
+ˆ
+S(j)
+c
+(ℓ+1) − R
+ˆ
+S(j)
+c
+(ℓ)
+�
+≤ 4C log(n ∨ m)
+ρ
+∆1(Du,−j).
+(C.19)
+Plugging the upper bound (C.18) and (C.19) into (C.15) and (C.14) respectively, we can guarantee
+Pu,−j
+�
+Rj > Q
+ˆ
+Sc∪{j}���Tj ≤ TU
+(ˆκ)
+�
+− Pu,−j
+�
+Rj > Q
+ˆ
+S(j)
+c
+∪{j}���Tj ≤ TU
+(ˆκ)
+�
+≤ 4C log(n ∨ m)
+ρTU\{j}
+(ˆκ(j)−Iu)
+�
+�12C log(n ∨ m)
+nTU\{j}
+(ˆκ(j)−Iu)
++
+2
+�
+TU\{j}
+(ˆκ(j)) − TU\{j}
+(ˆκ(j)−1)
+�
+TU\{j}
+(ˆκ(j))
+�
+� + 3(n ∨ m)−C
+TU\{j}
+(ˆκ(j))
+.
+C.3.3
+Proof of Lemma C.3
+Proof. Notice that
+Eu,−j
+�
+1
+| ˆS(j)
+c | + 1
+���Tj ≤ TU
+(ˆκ)
+�
+=
+n
+�
+s=0
+1
+s + 1
+Pu,−j
+�
+| ˆS(j)
+c | = s, Tj ≤ TU
+(ˆκ)
+�
+Pu,−j
+�
+Tj ≤ TU
+(ˆκ)
+�
+(i)
+≤
+n
+�
+s=0
+1
+s + 1
+Pu,−j
+�
+| ˆS(j)
+c | = s, Tj ≤ TU\{j}
+(ˆκ(j))
+�
+Pu,−j
+�
+Tj ≤ TU\{j}
+(ˆκ(j)−Iu)
+�
+(ii)
+=
+n
+�
+s=0
+1
+s + 1
+TU\{j}
+(ˆκ(j)) Eu,−j
+�
+1
+�
+| ˆS(j)
+c | = s
+��
+TU\{j}
+(ˆκ(j)−Iu)
+=
+TU\{j}
+(ˆκ(j))
+TU\{j}
+(ˆκ(j)−Iu)
+Eu,−j
+�
+1
+| ˆS(j)
+c | + 1
+�
+,
+(C.20)
+39
+
+where (i) holds due to TU
+(ˆκ) ≥ TU\{j}
+(ˆκ(j)−Iu) (c.f. Lemma A.3), and (ii) comes from the independence between Tj
+and ˆS(j)
+c
+such that
+Pu,−j
+�
+| ˆS(j)
+c | = s, Tj ≤ TU\{j}
+(ˆκ(j))
+�
+= Eu,−j
+�
+E
+�
+1
+�
+Tj ≤ TU\{j}
+(ˆκ(j))
+�
+1
+�
+| ˆS(j)
+c | = s
+� ��Du,−j, Dc
+��
+= TU\{j}
+(ˆκ(j)) Eu,−j
+�
+1
+�
+| ˆS(j)
+c | = s
+��
+.
+From the relation (C.16), we know
+Pu,−j
+�
+�| ˆS(j)
+c | ≤
+nTU\{j}
+(ˆκ(j))
+2
+�
+� ≤ (n ∨ m)−C.
+Together with (C.20), we have
+Eu,−j
+�
+1
+| ˆS(j)
+c | + 1
+���Tj ≤ TU
+(ˆκ)
+�
+≤
+TU\{j}
+(ˆκ(j))
+TU\{j}
+(ˆκ(j)−Iu)
+�
+�
+2
+nTU\{j}
+(ˆκ(j)) + 2
++ (n ∨ m)−C
+�
+� .
+C.4
+Calibration-assisted selection
+In this subsection, we provide the proofs for the results of calibration-assisted selection procedures, where the
+ranking threshold ˆκ depends on the test set Du and the calibration set Dc. From now on, we use E−(j,k)[·]
+and P−(j,k)[·] to denote the expectation and probability given Dc,−k and Du,−j.
+C.4.1
+Proof of Theorem 4
+Proof. From the definition of Q
+ˆ
+Sc∪{j} and Lemma A.1, we know that
+α −
+1
+| ˆSc| + 1
+≤
+1
+| ˆSc| + 1
+�
+i∈ ˆ
+Sc∪{j}
+1
+�
+Ri > Q
+ˆ
+Sc∪{j}�
+≤ α.
+(C.21)
+In addition, we also know {j ̸∈ PIj} = {Rj > Q
+ˆ
+Sc∪{j}}. Rearranging the right hand side of (C.21) gives
+1 {j ̸∈ PIj} = 1
+�
+Rj > Q
+ˆ
+Sc∪{j}�
+≤ α −
+1
+| ˆSc| + 1
+�
+i∈ ˆ
+Sc∪{j}
+1
+�
+Ri > Q
+ˆ
+Sc∪{j}�
++ 1
+�
+Rj > Q
+ˆ
+Sc∪{j}�
+= α −
+1
+| ˆSc| + 1
+�
+k∈ ˆ
+Sc
+�
+1
+�
+Rk > Q
+ˆ
+Sc∪{j}�
+− 1
+�
+Rj > Q
+ˆ
+Sc∪{j}��
+.
+(C.22)
+Given the dataset Du,−j, taking expectation on both sides of (C.22) conditional on the event Tj ≤ TU
+(ˆκ) yields
+Pu,−j
+�
+Yj ̸∈ PIj
+���Tj ≤ TU
+(ˆκ)
+�
+≤ α − Eu,−j
+�
+�
+1
+| ˆSc| + 1
+�
+k∈ ˆ
+Sc
+�
+1
+�
+Rk > Q
+ˆ
+Sc∪{j}�
+− 1
+�
+Rj > Q
+ˆ
+Sc∪{j}�� ���Tj ≤ TU
+(ˆκ)
+�
+� .
+(C.23)
+40
+
+Similarly, from the left hand side of (C.21), we can have
+Pu,−j
+�
+Yj ̸∈ PIj
+���Tj ≤ TU
+(ˆκ)
+�
+≥ α − Eu,−j
+�
+�
+1
+| ˆSc| + 1
+�
+k∈ ˆ
+Sc
+�
+1
+�
+Rk > Q
+ˆ
+Sc∪{j}�
+− 1
+�
+Rj > Q
+ˆ
+Sc∪{j}�� ���Tj ≤ TU
+(ˆκ)
+�
+�
+− Eu,−j
+�
+1
+| ˆSc| + 1
+���Tj ≤ TU
+(ˆκ)
+�
+.
+(C.24)
+For any j ∈ ˆSu and k ∈ C, we introduce a new selected virtual calibration set w.r.t. to (j, k),
+ˆS(j,k)
+c
+=
+�
+i ∈ C \ {k} : Ti ≤ TU\{j}
+ˆκ(j,k)
+�
+.
+According to Assumptions 2 and 4, we know
+ˆκ(j,k) = ˆκj←tu,k←tc ≥ ˆκj←tu = ˆκ(j).
+Together with Lemmas A.3 and A.2, we also know
+TU
+(ˆκ) ≤ TU\{j}
+(ˆκ(j)) ≤ TU\{j}
+(ˆκ(j,k)).
+(C.25)
+Hence we can guarantee ˆSc ⊆ ˆS(j,k)
+c
+∪ {k} if k ∈ ˆSc. Denote
+∆(j,k) = P−(j,k)
+�
+Rk > Q
+ˆ
+Sc∪{j}��Tj ≤ TU
+(ˆκ), Tk ≤ TU
+(ˆκ)
+�
+− P−(j,k)
+�
+Rj > Q
+ˆ
+Sc∪{j}��Tj ≤ TU
+(ˆκ), Tk ≤ TU
+(ˆκ)
+�
+=: ∆(j,k)
+k
+− ∆(j,k)
+j
+.
+(C.26)
+Notice that
+������
+Eu,−j
+�
+�
+1
+| ˆSc| + 1
+�
+k∈ ˆ
+Sc
+�
+1
+�
+Rk > Q
+ˆ
+Sc∪{j}�
+− 1
+�
+Rj > Q
+ˆ
+Sc∪{j}�� ��Tj ≤ TU
+(ˆκ)
+�
+�
+������
+=
+������
+Eu,−j
+�
+��
+k∈C
+1
+�
+k ∈ ˆSc
+�
+| ˆS(j,k)
+c
+| + 1
+�
+1
+�
+Rk > Q
+ˆ
+Sc∪{j}�
+− 1
+�
+Rj > Q
+ˆ
+Sc∪{j}�� ��Tj ≤ TU
+(ˆκ)
+�
+�
+������
++
+������
+Eu,−j
+�
+��
+k∈C
+| ˆSc| − | ˆS(j,k)
+c
+|
+| ˆS(j,k)
+c
+| + 1
+1
+�
+k ∈ ˆSc
+�
+| ˆSc| + 1
+�
+1
+�
+Rk > Q
+ˆ
+Sc∪{j}�
+− 1
+�
+Rj > Q
+ˆ
+Sc∪{j}�� ��Tj ≤ TU
+(ˆκ)
+�
+�
+������
+(i)
+≤ Eu,−j
+��
+k∈C
+P−(j,k)
+�
+k ∈ ˆSc|Tj ≤ TU
+(ˆκ)
+�
+��∆(j,k)��
+| ˆS(j,k)
+c
+| + 1
+�
++ Eu,−j
+�
+�max
+k∈C
+���| ˆSc| − | ˆS(j,k)
+c
+|
+���
+| ˆS(j,k)
+c
+| + 1
+�
+�
+= Eu,−j
+�
+�
+���∆(j,k)��� E−(j,k)
+�
+�
+�
+k∈C 1
+�
+k ∈ ˆSc
+�
+| ˆS(j,k)
+c
+| + 1
+���Tj ≤ TU
+(ˆκ)
+�
+�
+�
+� + Eu,−j
+�
+�max
+k∈C
+���| ˆSc| − | ˆS(j,k)
+c
+|
+���
+| ˆS(j,k)
+c
+| + 1
+�
+�
+(ii)
+≤ Eu,−j
+�
+max
+k∈C
+���∆(j,k)���
+�
++ Eu,−j
+�
+�max
+k∈C
+���| ˆSc| − | ˆS(j,k)
+c
+|
+���
+| ˆS(j,k)
+c
+| + 1
+�
+� ,
+(C.27)
+41
+
+where (i) follows from the tower rule and | ˆS(j,k)
+c
+| is independent of samples j and k, and (ii) holds since
+| ˆSc| ≤ | ˆS(j,k)
+c
+| + 1. Further, we can decompose ∆(j,k)
+j
+in (C.26) as
+∆(j,k)
+j
+=
+�
+P−(j,k)
+�
+Rj > Q
+ˆ
+Sc∪{j}��Tj ≤ TU
+(ˆκ), Tk ≤ TU
+(ˆκ)
+�
+− P−(j,k)
+�
+Rj > Q
+ˆ
+S(j,k)
+c
+∪{j,k}��Tj ≤ TU
+(ˆκ), Tk ≤ TU
+(ˆκ)
+� �
++
+�
+P−(j,k)
+�
+Rj > Q
+ˆ
+S(j,k)
+c
+∪{j,k}��Tj ≤ TU
+(ˆκ), Tk ≤ TU
+(ˆκ)
+�
+− P−(j,k)
+�
+Rj > Q
+ˆ
+S(j,k)
+c
+∪{j,k}��Tj ≤ TU\{j}
+(ˆκ(j,k)), Tk ≤ TU\{j}
+(ˆκ(j,k))
+� �
++ P−(j,k)
+�
+Rj > Q
+ˆ
+S(j,k)
+c
+∪{j,k}��Tj ≤ TU\{j}
+(ˆκ(j,k)), Tk ≤ TU\{j}
+(ˆκ(j,k))
+�
+=: ∆(j,k)
+j,1
++ ∆(j,k)
+j,2
++ P−(j,k)
+�
+Rj > Q
+ˆ
+S(j,k)
+c
+∪{j,k}��Tj ≤ TU\{j}
+(ˆκ(j,k)), Tk ≤ TU\{j}
+(ˆκ(j,k))
+�
+.
+(C.28)
+Similarly, for ∆(j,k)
+k
+in (C.26), we have
+∆(j,k)
+k
+=
+�
+P−(j,k)
+�
+Rk > Q
+ˆ
+Sc∪{j}��Tj ≤ TU
+(ˆκ), Tk ≤ TU
+(ˆκ)
+�
+− P−(j,k)
+�
+Rk > Q
+ˆ
+S(j,k)
+c
+∪{j,k}��Tj ≤ TU
+(ˆκ), Tk ≤ TU
+(ˆκ)
+� �
++
+�
+P−(j,k)
+�
+Rk > Q
+ˆ
+S(j,k)
+c
+∪{j,k}��Tj ≤ TU
+(ˆκ), Tk ≤ TU
+(ˆκ)
+�
+− P−(j,k)
+�
+Rk > Q
+ˆ
+S(j,k)
+c
+∪{j,k}��Tj ≤ TU\{j}
+(ˆκ(j,k)), Tk ≤ TU\{j}
+(ˆκ(j,k))
+� �
++ P−(j,k)
+�
+Rk > Q
+ˆ
+S(j,k)
+c
+∪{j,k}��Tj ≤ TU\{j}
+(ˆκ(j,k)), Tk ≤ TU\{j}
+(ˆκ(j,k))
+�
+=: ∆(j,k)
+k,1
++ ∆(j,k)
+k,2
++ P−(j,k)
+�
+Rk > Q
+ˆ
+S(j,k)
+c
+∪{j,k}��Tj ≤ TU\{j}
+(ˆκ(j,k)), Tk ≤ TU\{j}
+(ˆκ(j,k))
+�
+.
+(C.29)
+Using the identical distribution of sample j and k and the fact ˆS(j,k)
+c
+, TU\{j}
+(ˆκ(j,k)) are independent of sample j
+and k, we have
+P−(j,k)
+�
+Rk > Q
+ˆ
+S(j,k)
+c
+∪{j,k}��Tj ≤ TU\{j}
+(ˆκ(j,k)), Tk ≤ TU\{j}
+(ˆκ(j,k))
+�
+=P−(j,k)
+�
+Rj > Q
+ˆ
+S(j,k)
+c
+∪{j,k}��Tj ≤ TU\{j}
+(ˆκ(j,k)), Tk ≤ TU\{j}
+(ˆκ(j,k))
+�
+.
+Taking account of (C.26), (C.28) and (C.29), the equality above results in
+∆(j,k) = ∆(j,k)
+k,1
++ ∆(j,k)
+k,2
+−
+�
+∆(j,k)
+j,1
++ ∆(j,k)
+j,2
+�
+.
+(C.30)
+Since ˆSc \ {k} ⊆ ˆS(j,k)
+c
+, under the event j ∈ ˆSu, we know that
+| ˆS(j,k)
+c
+| − | ˆSc| + 1 ≤
+�
+i∈ ˆ
+S(j,k)
+c
+1
+�
+TU
+(ˆκ) < Ti ≤ TU\{j}
+(ˆκ(j,k))
+�
+=
+�
+i∈ ˆ
+S(j,k)
+c
+1
+�
+TU
+(ˆκ(j)) < Ti ≤ TU\{j}
+(ˆκ(j,k))
+�
+≤
+�
+i∈ ˆ
+S(j,k)
+c
+1
+�
+TU\{j}
+(ˆκ(j,k)−Ic−1) < Ti ≤ TU\{j}
+(ˆκ(j,k))
+�
+=: d(j,k),
+(C.31)
+where the last inequality holds since ˆκ(j,k) ≤ ˆκ(j) + Ic.
+42
+
+Bound ∆(j,k)
+j,1
+and ∆(j,k)
+k,1 .
+We first bound the rank of value Q
+ˆ
+Sc∪{j} in the set {Ri : i ∈ ˆS(j,k)
+c
+}. For k ∈ ˆSc
+(that is, Tk ≤ TU
+(ˆκ)), we have
+�
+i∈ ˆ
+S(j,k)
+c
+1
+�
+Ri ≤ Q
+ˆ
+Sc∪{j}�
+≥
+�
+i∈ ˆ
+S(j,k)
+c
+∪{j,k}
+1
+�
+Ri ≤ Q
+ˆ
+Sc∪{j}�
+− 2
+(i)
+≥
+�
+i∈ ˆ
+Sc∪{j}
+1
+�
+Ri ≤ Q
+ˆ
+Sc∪{j}�
+− 2
+= ⌈(1 − α)(1 + | ˆSc|)⌉ − 2
+(ii)
+≥ ⌈(1 − α)(| ˆS(j,k)
+c
+| − d(j,k) + 2)⌉ − 2,
+(C.32)
+and
+�
+i∈ ˆ
+S(j,k)
+c
+1
+�
+Ri ≤ Q
+ˆ
+Sc∪{j}�
+≤
+�
+i∈ ˆ
+S(j,k)
+c
+∪{j,k}
+1
+�
+Ri ≤ Q
+ˆ
+Sc∪{j}�
+(iii)
+≤
+��
+i∈ ˆ
+Sc∪{j}
+1
+�
+Ri ≤ Q
+ˆ
+Sc∪{j}�
++ d(j,k) − 1
+≤ ⌈(1 − α)(2 + | ˆS(j,k)
+c
+|)⌉ + d(j,k) − 1,
+(C.33)
+where (i), (ii) holds due to ˆSc ∪ {j} ⊆ ˆS(j,k)
+c
+∪ {j, k}, and (ii), (iii) comes from (C.31). Similarly, we have
+�
+i∈ ˆ
+S(j,k)
+c
+1
+�
+Ri ≤ Q
+ˆ
+S(j,k)
+c
+∪{j,k}�
+≥
+�
+i∈ ˆ
+S(j,k)
+c
+∪{j,k}
+1
+�
+Ri ≤ Q
+ˆ
+S(j,k)
+c
+∪{j,k}�
+− 2
+= ⌈(1 − α)(2 + | ˆS(j,k)
+c
+|)⌉ − 2,
+(C.34)
+and
+�
+i∈ ˆ
+S(j,k)
+c
+1
+�
+Ri ≤ Q
+ˆ
+S(j,k)
+c
+∪{j,k}�
+≤
+�
+i∈ ˆ
+S(j,k)
+c
+∪{j,k}
+1
+�
+Ri ≤ Q
+ˆ
+S(j,k)
+c
+∪{j,k}�
+= ⌈(1 − α)(2 + | ˆS(j,k)
+c
+|)⌉.
+(C.35)
+Combining (C.32)-(C.35), we can guarantee
+���Q
+ˆ
+Sc∪{j} − Q
+ˆ
+S(j,k)
+c
+∪{j,k}��� ≤ max
+�
+R
+ˆ
+S(j,k)
+c
+(⌈(1−α)(2+| ˆ
+S(j,k)
+c
+|)⌉+d(j,k)−1) − R
+ˆ
+S(j,k)
+c
+(⌈(1−α)(2+| ˆ
+S(j,k)
+c
+|)⌉−2),
+R
+ˆ
+S(j,k)
+c
+(⌈(1−α)(2+| ˆ
+S(j,k)
+c
+|)⌉) − R
+ˆ
+S(j,k)
+c
+(⌈(1−α)(| ˆ
+S(j,k)
+c
+|−d(j,k)+2)⌉−2)
+�
+≤ R
+ˆ
+S(j,k)
+c
+(⌈(1−α)(| ˆ
+S(j,k)
+c
+|+2)⌉+d(j,k)) − R
+ˆ
+S(j,k)
+c
+(⌈(1−α)(| ˆ
+S(j,k)
+c
+|+2−d(j,k))⌉−2)
+=: R
+ˆ
+S(j,k)
+c
+(U (j,k)) − R
+ˆ
+S(j,k)
+c
+(L(j,k)).
+(C.36)
+In addition, using Assumptions 2 and 4, we know for any j ∈ U, it holds that
+TU
+(ˆκ) ≥ TU\{j}
+(ˆκ(j)−Iu) ≥ TU\{j}
+(ˆκ(j,k)−Iu−Ic).
+(C.37)
+43
+
+Since the samples j and k are independent of R
+ˆ
+S(j,k)
+c
+(U (j,k)) and R
+ˆ
+S(j,k)
+c
+(L(j,k)), we have
+|∆(j,k)
+k,1 | ≤ E−(j,k)
+����1
+�
+Rk > Q
+ˆ
+Sc∪{j}�
+− 1
+�
+Rk > Q
+ˆ
+S(j,k)
+c
+∪{j,k}����
+��Tj ≤ TU
+(ˆκ), Tk ≤ TU
+(ˆκ)
+�
+(i)
+≤ P−(j,k)
+�
+R
+ˆ
+S(j,k)
+c
+(L(j,k)) < Rk ≤ R
+ˆ
+S(j,k)
+c
+(U (j,k))
+��Tj ≤ TU
+(ˆκ), Tk ≤ TU
+(ˆκ)
+�
+(ii)
+≤
+P−(j,k)
+�
+R
+ˆ
+S(j,k)
+c
+(L(j,k)) < Rk ≤ R
+ˆ
+S(j,k)
+c
+(U (j,k)), Tj ≤ TU\{j}
+(ˆκ(j,k))
+�
+P−(j,k)
+�
+Tj ≤ TU\{j}
+(ˆκ(j,k)−Iu−Ic), Tk ≤ TU\{j}
+(ˆκ(j,k)−Iu−Ic)
+�
+≤
+TU\{j}
+(ˆκ(j,k))
+�
+R
+ˆ
+S(j,k)
+c
+(U (j,k)) − R
+ˆ
+S(j,k)
+c
+(L(j,k))
+�
+�
+TU\{j}
+(ˆκ(j,k)−Iu−Ic)
+�2
+,
+(C.38)
+where (i) comes from (C.36) and (ii) holds due to (C.37). Similarly, we can also have
+|∆(j,k)
+j,1 | ≤ E−(j,k)
+����1
+�
+Rj > Q
+ˆ
+Sc∪{j}�
+− 1
+�
+Rj > Q
+ˆ
+S(j,k)
+c
+∪{j,k}����
+��Tj ≤ TU
+(ˆκ), Tk ≤ TU
+(ˆκ)
+�
+≤ P−(j,k)
+�
+R
+ˆ
+S(j,k)
+c
+(L(j,k)) < Rj ≤ R
+ˆ
+S(j,k)
+c
+(U (j,k))
+��Tj ≤ TU
+(ˆκ), Tk ≤ TU
+(ˆκ)
+�
+≤
+P−(j,k)
+�
+R
+ˆ
+S(j,k)
+c
+(L(j,k)) < Rj ≤ R
+ˆ
+S(j,k)
+c
+(U (j,k)), Tk ≤ TU\{j}
+(ˆκ(j,k))
+�
+P−(j,k)
+�
+Tj ≤ TU\{j}
+(ˆκ(j,k)−Iu−Ic), Tk ≤ TU\{j}
+(ˆκ(j,k)−Iu−Ic)
+�
+≤
+TU\{j}
+(ˆκ(j,k))
+�
+R
+ˆ
+S(j,k)
+c
+(U (j,k)) − R
+ˆ
+S(j,k)
+c
+(L(j,k))
+�
+�
+TU\{j}
+(ˆκ(j,k)−Iu−Ic)
+�2
+.
+(C.39)
+Bound ∆(j,k)
+j,2
+and ∆(j,k)
+k,2 .
+For ∆(j,k)
+j,2 , we have the following upper bound
+∆(j,k)
+j,2
+(i)
+≤
+P−(j,k)
+�
+Rj > Q
+ˆ
+S(j,k)
+c
+∪{j,k}, Tj ≤ TU\{j}
+(ˆκ(j,k)), Tk ≤ TU\{j}
+(ˆκ(j,k))
+�
+P−(j,k)
+�
+Tj ≤ TU
+(ˆκ), Tk ≤ TU
+(ˆκ)
+�
+−
+P−(j,k)
+�
+Rj > Q
+ˆ
+S(j,k)
+c
+∪{j,k}, Tj ≤ TU\{j}
+(ˆκ(j,k)), Tk ≤ TU\{j}
+(ˆκ(j,k))
+�
+P−(j,k)
+�
+Tj ≤ TU\{j}
+(ˆκ(j,k)), Tk ≤ TU\{j}
+(ˆκ(j,k))
+�
+(ii)
+≤ 1 −
+P−(j,k)
+�
+Tj ≤ TU
+(ˆκ), Tk ≤ TU
+(ˆκ)
+�
+P−(j,k)
+�
+Tj ≤ TU\{j}
+(ˆκ(j,k)), Tk ≤ TU\{j}
+(ˆκ(j,k))
+�
+=
+P−(j,k)
+�
+Tj ≤ TU\{j}
+(ˆκ(j,k)), TU
+(ˆκ) < Tk ≤ TU\{j}
+(ˆκ(j,k))
+�
+P−(j,k)
+�
+Tj ≤ TU\{j}
+(ˆκ(j,k)), Tk ≤ TU\{j}
+(ˆκ(j,k))
+�
++
+P−(j,k)
+�
+TU
+(ˆκ) < Tj ≤ TU\{j}
+(ˆκ(j,k)), Tk ≤ TU
+(ˆκ)
+�
+P−(j,k)
+�
+Tj ≤ TU\{j}
+(ˆκ(j,k)), Tk ≤ TU\{j}
+(ˆκ(j,k))
+�
+(iii)
+≤
+2
+�
+TU\{j}
+(ˆκ(j,k)) − TU\{j}
+(ˆκ(j,k)−Iu−Ic)
+�
+TU\{j}
+(ˆκ(j))
+,
+(C.40)
+44
+
+where (i) and (ii) holds since TU\{j}
+(ˆκ(j,k)) ≥ TU
+(ˆκ) in (C.25), and (iii) comes from (C.37) and the towerrule.
+Similarly, we can get the lower bound of ∆(j,k)
+j,2
+as
+∆(j,k)
+j,2
+≥
+P−(j,k)
+�
+Rj > Q
+ˆ
+S(j,k)
+c
+∪{j,k}, Tj ≤ TU
+(ˆκ), Tk ≤ TU
+(ˆκ)
+�
+P−(j,k)
+�
+Tj ≤ TU\{j}
+(ˆκ(j,k)), Tk ≤ TU\{j}
+(ˆκ(j,k))
+�
+−
+P−(j,k)
+�
+Rj > Q
+ˆ
+S(j,k)
+c
+∪{j,k}, Tj ≤ TU\{j}
+(ˆκ(j,k)), Tk ≤ TU\{j}
+(ˆκ(j,k))
+�
+P−(j,k)
+�
+Tj ≤ TU\{j}
+(ˆκ(j,k)), Tk ≤ TU\{j}
+(ˆκ(j,k))
+�
+≥ −
+P−(j,k)
+�
+TU
+(ˆκ) < Tj ≤ TU\{j}
+(ˆκ(j,k)), Tk ≤ TU
+(ˆκ)
+�
+P−(j,k)
+�
+Tj ≤ TU\{j}
+(ˆκ(j,k)), Tk ≤ TU\{j}
+(ˆκ(j,k))
+�
+−
+P−(j,k)
+�
+Tj ≤ TU\{j}
+(ˆκ(j,k)), TU
+(ˆκ) < Tk ≤ TU\{j}
+(ˆκ(j,k))
+�
+P−(j,k)
+�
+Tj ≤ TU\{j}
+(ˆκ(j,k)), Tk ≤ TU\{j}
+(ˆκ(j,k))
+�
+≥ −
+2
+�
+TU\{j}
+(ˆκ(j,k)) − TU\{j}
+(ˆκ(j,k)−Iu−Ic)
+�
+TU\{j}
+(ˆκ(j))
+.
+(C.41)
+Applying the same calculations in (C.40) and (C.41) to ∆(j,k)
+k,2 , we can get
+|∆(j,k)
+j,2 |, |∆(j,k)
+k,2 | ≤
+2
+�
+TU\{j}
+(ˆκ(j,k)) − TU\{j}
+(ˆκ(j,k)−Iu−Ic)
+�
+TU\{j}
+(ˆκ(j))
+.
+(C.42)
+Conclusion.
+Substituting (C.39), (C.38) and (C.42) into (C.30), we have
+|∆(j,k)| ≤
+2TU\{j}
+(ˆκ(j,k))
+�
+R
+ˆ
+S(j,k)
+c
+(U (j,k)) − R
+ˆ
+S(j,k)
+c
+(L(j,k))
+�
+�
+TU\{j}
+(ˆκ(j,k)−Iu−Ic)
+�2
++
+4
+�
+TU\{j}
+(ˆκ(j,k)) − TU\{j}
+(ˆκ(j,k)−Iu−Ic)
+�
+TU\{j}
+(ˆκ(j))
+,
+where U (j,k) = ⌈(1 − α)(| ˆS(j,k)
+c
+| + 2)⌉ + d(j,k) and L(j,k) = ⌈(1 − α)(| ˆS(j,k)
+c
+| + 2 − d(j,k))⌉ − 2. In addition, from
+(C.31), we know
+���| ˆSc| − | ˆS(j,k)
+c
+|
+���
+| ˆS(j,k)
+c
+| + 1
+≤
+d(j,k)
+| ˆS(j,k)
+c
+| + 1
+.
+Plugging two upper bounds above into (C.27) can finish the proof.
+C.4.2
+Proof of Theorem 5
+Proof of Theorem 5. Denote ˆSc(κ) =
+�
+i ∈ C \ {k} : Ti ≤ TU\{j}
+(κ)
+�
+. From the definition of ˆS(j,k)
+c
+, we can write
+ˆS(j,k)
+c
+= ˆSc(ˆκ(j,k)). In addition, we write
+d(j,k) =
+�
+i∈ ˆ
+S(j,k)
+c
+1
+�
+TU\{j}
+(ˆκ(j,k)−Ic−1) < Ti ≤ TU\{j}
+(ˆκ(j,k))
+�
+=: d(j,k)(ˆκ(j,k)).
+45
+
+Then we have
+R
+ˆ
+S(j,k)
+c
+(U (j,k)) − R
+ˆ
+S(j,k)
+c
+(L(j,k)) =
+U (j,k)−1
+�
+ℓ=L(j,k)
+R
+ˆ
+S(j,k)
+c
+(ℓ+1) − R
+ˆ
+S(j,k)
+c
+(ℓ)
+≤
+�
+d(j,k) + 2
+�
+max
+1≤ℓ≤| ˆ
+S(j,k)
+c
+|−1
+�
+R
+ˆ
+S(j,k)
+c
+(ℓ+1) − R
+ˆ
+S(j,k)
+c
+(ℓ)
+�
+=
+�
+d(j,k)(ˆκ(j,k)) + 2
+�
+max
+1≤ℓ≤| ˆ
+Sc(ˆκ(j,k))|−1
+�
+R
+ˆ
+Sc(ˆκ(j,k))
+(ℓ+1)
+− R
+ˆ
+Sc(ˆκ(j,k))
+(ℓ)
+�
+≤
+max
+⌈γm⌉≤κ≤m
+��
+d(j,k)(κ) + 2
+�
+max
+1≤ℓ≤| ˆ
+Sc(κ)|−1
+�
+R
+ˆ
+Sc(κ)
+(ℓ+1) − R
+ˆ
+Sc(κ)
+(ℓ)
+��
+.
+(C.43)
+Given Du,−j, applying Lemma A.6 with t1 = TU\{j}
+(κ−Ic−1) and t2 = TU\{j}
+(κ)
+, we can have
+d(j,k)(κ) ≤ | ˆSc(κ)|
+�
+�TU\{j}
+(κ)
+− TU\{j}
+(κ−Ic−1)
+TU\{j}
+(κ)
++ 6C log(n ∨ m)
+| ˆSc(κ)|
+�
+� ,
+(C.44)
+holds with probability at least 1 − (n ∨ m)−C. In addition, given Du,−j, applying Lemma A.5, we get
+max
+1≤ℓ≤| ˆ
+Sc(κ)|−1
+�
+R
+ˆ
+Sc(κ)
+(ℓ+1) − R
+ˆ
+Sc(κ)
+(ℓ)
+�
+≤ 1
+ρ
+1
+1 − 2
+�
+C log(n∨m)
+| ˆ
+Sc(κ)|+1
+2C log(n ∨ m)
+| ˆSc(κ)| + 1
+,
+(C.45)
+holds with probability at least 1 − (n ∨ m)−C. Substituting (C.44) and (C.45) into (C.43) and using union
+bound, with probability at least 1 − (n ∨ m)−C+2, we have
+R
+ˆ
+S(j,k)
+c
+(U (j,k)) − R
+ˆ
+S(j,k)
+c
+(L(j,k)) ≤
+max
+⌈γm⌉≤κ≤m
+�
+�
+�
+1
+ρ
+2C log(n ∨ m)
+1 − 2
+�
+C log(n∨m)
+| ˆ
+Sc(κ)|+1
+�
+�TU\{j}
+(κ)
+− TU\{j}
+(κ−Ic−1)
+TU\{j}
+(κ)
++ 6C log(n ∨ m)
+| ˆSc(κ)|
+�
+�
+�
+�
+�
+(i)
+≤
+max
+⌈γm⌉≤κ≤m
+�
+�
+�
+�
+�
+�
+�
+1
+ρ
+2C log(n ∨ m)
+1 − 2
+�
+C log(n∨m)
+nTU\{j}
+(κ)
+/2+1
+�
+�TU\{j}
+(κ)
+− TU\{j}
+(κ−Ic−1)
+TU\{j}
+(κ)
++ 6C log(n ∨ m)
+nTU\{j}
+(κ)
+/2
+�
+�
+�
+�
+�
+�
+�
+�
+�
+(ii)
+≤ 4C log(n ∨ m)
+ρ
+�
+�
+max⌈γm⌉≤κ≤m
+�
+TU\{j}
+(κ)
+− TU\{j}
+(κ−Ic−1)
+�
+TU\{j}
+(κ)
++ 6C log(n ∨ m)
+nTU\{j}
+(κ)
+/2
+�
+�
+(iii)
+≤ 4C log(n ∨ m)
+ρTU\{j}
+(⌈γm⌉)
+�4CIc log(n ∨ m)
+m + 1
++ 12C log(n ∨ m)
+n
+�
+,
+(C.46)
+where (i) follows from Lemma A.7, (ii) comes from Lemma A.8, and (iii) follows from Lemma A.4. Using
+Lemma A.4 again, we have
+TU\{j}
+(ˆκ(j,k)) − TU\{j}
+(ˆκ(j,k)−Iu−Ic) ≤ (Ic + Iu)
+max
+1≤ℓ≤m−2
+�
+TU\{j}
+(ℓ+1) − TU\{j}
+(ℓ)
+�
+≤ 4C(Ic + Iu) log(n ∨ m)
+m
+,
+(C.47)
+46
+
+holds with probability at least 1 − 2(n ∨ m)−C. With probability at least 1 − (n ∨ m)−C+2, we can guarantee
+that
+d(j,k)
+| ˆS(j,k)
+c
+| + 1
+(i)
+≤
+max
+⌈γm⌉≤κ≤m
+�
+d(j,k)(κ)
+| ˆSc(κ)| + 1
+�
+≤
+max
+⌈γm⌉≤κ≤m
+�
+�
+�
+TU\{j}
+(κ)
+− TU\{j}
+(κ−Ic−1)
+TU\{j}
+(κ)
++ 6C log(n ∨ m)
+| ˆSc(κ)|
+�
+�
+�
+(ii)
+≤
+max
+⌈γm⌉≤κ≤m
+�
+�
+�
+TU\{j}
+(κ)
+− TU\{j}
+(κ−Ic−1)
+TU\{j}
+(κ)
++ 6C log(n ∨ m)
+nTU\{j}
+(κ)
+/2
+�
+�
+�
+≤
+max⌈γm⌉≤κ≤m
+�
+TU\{j}
+(κ)
+− TU\{j}
+(κ−Ic−1)
+�
+TU\{j}
+(⌈γm⌉)
++ 12C log(n ∨ m)
+nTU\{j}
+(⌈γm⌉)
+(iii)
+≤ 8CIc log(n ∨ m)
+γm
++ 24C log(n ∨ m)
+nγ
+,
+(C.48)
+where (i) holds due to (C.44), (ii) comes from Lemma A.7, and (iii) comes from Lemma A.4 and A.8. Plugging
+(C.46), (C.47) and (C.48) into (11), we can get
+∆(−(j,k)) ≲
+log2(n ∨ m)
+ρ(κ − (Ic + Iu)/m)2
+�Ic + Iu
+m + 1 + 1
+n
+�
+,
+where we also used the fact TU\{j}
+(⌈γm⌉) ≤ TU\{j}
+(ˆκ(j,k)) since ˆκ(j,k) ≥ ˆκ ≥ ⌈γm⌉. Then the conclusion follows from
+Lemma 1 immediately.
+C.5
+Prediction-oriented selection with conformal p-values
+In section 3.3, we split the calibration set according the null hypothesis and alternative hypothesis, that is
+C = C0 ∪ C1. Denote n0 = |C0| and n1 = |C1|. Let us first recall the definition of conformal p-value:
+pj = 1 + �
+i∈C0 1 {Ti ≤ Tj}
+n0 + 1
+.
+Therefore, the ranking threshold ˆκ only depends on the test set Du and the null calibration set Dc0. The
+selected calibration set is defined as
+ˆSc1 =
+�
+i ∈ C1 : Ti ≤ TU
+(ˆκ)
+�
+.
+In the following proof, we will fix the null calibration set Dc0 = {(Xi, Yi) : i ∈ C, Yi ≤ b0}, and then the
+randomness of ˆκ only comes from the test set. Moreover, the index set C1 is also fixed once given Dc0.
+C.5.1
+Proof of Proposition 3.1
+Proof. Notice that, there may exist ties in {pj : j ∈ U}, but there is no tie in {Tj : j ∈ U}. Let Rank(pj) =
+�
+i∈U 1 {pi ≤ pj} and Rank(Tj) = �
+i∈U 1 {Ti ≤ Tj} be the ranks of pj and Tj in the test set, respectively.
+47
+
+Then we have Rank(pj) ≥ Rank(Tj), which means
+�
+pj ≤ pU
+(ˆκ)
+�
+= {Rank(pj) ≤ ˆκ} =⇒ {Rank(Tj) ≤ ˆκ} =
+�
+Tj ≤ TU
+(ˆκ)
+�
+.
+By the definition of ˆκ = max
+�
+τ : pU
+(τ) ≤ δ(τ)
+�
+, we know
+�
+Tj ≤ TU
+(ˆκ)
+�
+=
+�
+Tj ≤ max
+�
+Ti : pi = pU
+(ˆκ)
+��
+=⇒
+�
+pj ≤ pU
+(ˆκ)
+�
+.
+Therefore, we have proved that {p(Xj) ≤ pU
+(τ)} = {Tj ≤ TU
+(τ)} for any j ∈ U.
+C.5.2
+Proof of Proposition 3.2
+Proof. Conclusion 1.
+For any j ∈ ˆSu, the Lemma 1 in Fithian and Lei (2020) has showed ˆκ = ˆκ(j)
+for the step-up procedures. Next, we prove the conclusion for j ∈ U \ ˆSu. Denote ri as the rank of Ti
+in the set TU. Denote r(j)
+i
+as the rank of Ti in the set TU but the value of Tj is replaced by 0, that is
+{0, T1, ..., Tj−1, Tj+1, ..., Tm}. Then we know
+r(j)
+i
+= ri + 1 for ri < rj; and r(j)
+i
+= ri for ri > rj.
+(C.49)
+Recall the definition of ˆκ in step-up procedures, ˆκ = max{r :
+pU
+(r) ≤ δ(r)}. Together with (C.49) and
+ˆκ ≥ ⌈γm⌉ in Assumption 3, we have
+ˆκ(j) − ˆκ =
+max
+ˆκ+1≤r≤rj
+�
+r : δ(r) < pU
+(r) ≤ δ(r + 1)
+�
+− ˆκ
+=
+rj
+�
+r=ˆκ+1
+1
+�
+δ(r) < pU
+(r) ≤ δ(r + 1)
+�
+≤
+max
+⌈γm⌉≤r≤m−1
+m
+�
+i=1
+1 {δ(r) < pi ≤ δ(r + 1)} .
+(C.50)
+Given the calibration set, we know {pi : i ∈ U} are i.i.d. random variables with
+PDc
+�
+pi =
+k
+n0 + 1
+�
+= PDc
+� �
+k∈C0
+1 {Tk ≤ Ti} = k − 1
+�
+= PDc
+�
+TC0
+(k−1) < Ti ≤ TC0
+(k)
+�
+= TC0
+(k) − TC0
+(k−1),
+where TC0
+(k) is the k-th smallest value in TC0. Now we denote Ω(r) = {k ∈ [n0] : δ(r) <
+k
+n0+1 < δ(r + 1)}.
+Then we have
+PDc (δ(r) < pi ≤ δ(r + 1)) =
+�
+k∈Ω(r)
+PDc
+�
+pi =
+k
+n0 + 1
+�
+=
+�
+k∈Ω(r)
+TC0
+(k) − TC0
+(k−1) =: ω(r).
+(C.51)
+Similar to Lemma A.6, given Dc, we can verify that for any C ≥ 1,
+�����
+1
+m
+m
+�
+i=1
+1 {δ(r) < pi ≤ δ(r + 1)} − ω(r)
+����� ≤ 2eC log m
+m
++ 2
+�
+eCω(r) log m
+m
+,
+(C.52)
+48
+
+holds with probability at least 1 − (n ∨ m)−C. Invoking maximal spacing in Lemma A.4, we have
+P
+�
+ω(r) ≥ min
+�2C|Ω(r)| log m
+n0 + 1
+, 1
+��
+≤ m−C.
+(C.53)
+Combining (C.50)-(C.53), we can guarantee that
+ˆκ(j) − ˆκ ≤ 4eC log m + 4m
+max
+1≤r≤m−1 ω(r)
+≤ 4eC log m + 4m
+max
+1≤r≤m−1
+��2C|Ω(r)| log(n ∨ m)
+n0 + 1
+�
+∧ 1
+�
+,
+holds with probability at least 1 − 2m−C+1.
+Conclusion 2. If we replace Tk with 1 for any k ∈ C, the corresponding p-value is
+p(k)
+i
+=
+1 + �
+l∈C0\{k} 1 {Tl ≤ Ti}
+n0 + 1
+,
+for i ∈ U.
+It indicates that 0 ≤ pi − p(k)
+i
+≤ 1/(n0 + 1) for any i ∈ U. Hence we have
+ˆκ(k) − ˆκ =
+max
+ˆκ+1≤r≤m
+�
+r : δ(r) < pU
+(r) ≤ δ(r) +
+1
+n0 + 1
+�
+− ˆκ
+≤
+max
+1≤r≤m−1
+m
+�
+i=1
+1
+�
+δ(r) < pi ≤ δ(r) +
+1
+n0 + 1
+�
+.
+Notice that
+����
+�
+k ∈ [n0] : δ(r) < k + 1
+n0 + 1 ≤ δ(r) +
+1
+n0 + 1
+����� ≤ 1.
+Similar arguments yield that
+ˆκ(k) − ˆκ ≤ 4eC log m + 8Cm log m
+n0 + 1
+,
+holds with probability at least 1 − 2m−C+1.
+D
+Proof of Auxiliary Lemmas
+D.1
+Proof of Lemma A.3
+Proof. Let Rank(Ti) for i ∈ U be the rank of Ti in the test set {Ti : i ∈ U}. Notice that, ˆκ(j) = ˆκ if
+Rank(Tj) ≤ ˆκ, and ˆκ(j) ≤ ˆκ + Iu if Rank(Tj) > ˆκ. Next we discuss the value of TU\{j}
+(ˆκ(j)) in different scenarios:
+• If Rank(Tj) > ˆκ, then TU\{j}
+(ˆκ(j)−Iu) ≤ TU\{j}
+(ˆκ)
+= TU
+(ˆκ).
+• If Rank(Tj) = ˆκ, then TU\{j}
+(ˆκ(j)) = TU\{j}
+(ˆκ)
+= TU
+(ˆκ+1) and TU\{j}
+(ˆκ(j)−1) = TU\{j}
+(ˆκ−1) = TU
+(ˆκ−1).
+• If Rank(Tj) < ˆκ, then TU\{j}
+(ˆκ(j)) = TU\{j}
+(ˆκ)
+= TU
+(ˆκ+1) and TU\{j}
+(ˆκ(j)−1) = TU\{j}
+(ˆκ−1) = TU
+(ˆκ).
+Then the conclusion follows immediately.
+49
+
+D.2
+Proof of Lemma A.4
+Lemma D.1 (Representation of spacing (Arnold et al., 2008)). Let U1, · · · , Un
+i.i.d.
+∼ Unif([0, 1]), and U(1) ≤
+U(2) ≤ · · · ≤ U(n) be their order statistics. Then
+�
+U(1) − U(0), · · · , U(n+1) − U(n)
+� d=
+�
+V1
+�n+1
+k=1 Vk
+, · · · ,
+Vn+1
+�n+1
+k=1 Vk
+�
+,
+where U0 = 0, U(n+1) = 1, and V1, · · · , Vn+1
+i.i.d.
+∼ Exp(1).
+Lemma D.2 (Quantile transformation of order statistics, Theorem 1.2.5 in Reiss (2012)). Let X1, · · · , Xn be
+i.i.d. random variables with CDF F(·), and U1, · · · , Un
+i.i.d.
+∼ Unif([0, 1]), then
+�
+F −1(U(1)), · · · , F −1(U(n))
+� d=
+�
+X(1), · · · , X(n)
+�
+.
+Fact 1. For the random variable X ∼ Exp(λ), it holds that P (X ≥ x) = e−λx.
+Fact 2. For the random variable X ∼ χ2
+ν, it holds that P (X − ν ≥ x) ≤ e−νx2/8.
+Proof of Lemma A.4. Using the spacing representation in Lemma D.1, we have
+PDu
+�
+� max
+0≤ℓ≤n
+�
+U(ℓ+1) − U(ℓ)
+�
+≥
+1
+1 − 2
+�
+C log(n∨m)
+n+1
+C log(n ∨ m)
+n + 1
+�
+�
+= P
+�
+� max
+0≤ℓ≤n
+Vℓ
+�n+1
+i=1 Vi
+≥
+1
+1 − 2
+�
+C log(n∨m)
+n+1
+2C log(n ∨ m)
+n + 1
+�
+�
+≤ P
+�
+max
+0≤ℓ≤n Vℓ ≥ 2C log(n ∨ m)
+�
++ P
+�
+1
+n + 1
+n+1
+�
+i=1
+Vi ≤ 1 − 2
+�
+C log(n ∨ m)
+n + 1
+�
+,
+(D.1)
+where U(0) = 0 and U(n+1) = 1. By the tail probability of Exp(1) in Fact 1 and union bound, we know
+P
+�
+max
+0≤ℓ≤n Vℓ ≥ 2C log(n ∨ m)
+�
+≤ (n + 1) P (V1 ≥ 2C log(n ∨ m))
+= (n + 1) (n ∨ m)−2C ≤ (n ∨ m)−C,
+(D.2)
+holds for any C ≥ 1. In addition, we know that 1
+2
+�n+1
+i=1 Vi ∼ Γ(n + 1, 2)
+d= χ2
+2(n+1). Using the tail bound of
+χ2 distribution with ν = 2(n + 1) in Fact 2, we have
+P
+�
+1
+n + 1
+n+1
+�
+i=1
+Vi ≤ 1 − 2
+�
+C log(n ∨ m)
+n + 1
+�
+≤ (n ∨ m)−C.
+(D.3)
+Substituting (D.2) and (D.3) into (D.1) gives the desired result.
+D.3
+Proof of Lemma A.5
+Proof. Notice that, for any Sc ⊆ C, we can write
+�
+ˆSc(t) = Sc
+�
+=
+� �
+i∈Sc
+{Ti ≤ t}
+� �
+�
+� �
+i∈C\Sc
+{Ti > t}
+�
+� .
+(D.4)
+50
+
+Then for any i ∈ Sc, we have
+P
+�
+Ri ≤ r
+�� ˆSc(t) = Sc
+�
+= P
+�
+Ri ≤ r
+��Ti ≤ t
+�
+= F(R,T )(r, t)
+t
+=: G(r).
+Hence, given ˆSc(t) = Sc ⊆ C, {Ri}i∈Sc are i.i.d. random variables with the common CDF G(·). Applying
+Lemma D.2, we know there exist Ui
+i.i.d.
+∼ Unif([0, 1]) for i ∈ Sc such that
+�
+RSc
+(1), · · · , RSc
+(|Sc|)| ˆSc(t) = Sc
+� d=
+�
+G−1(U(1), · · · , G−1(U(|Sc|)
+�
+.
+(D.5)
+Let G−1(·) be the inverse function of G(·), and use our assumption on
+d
+drF(R,T )(r, t) ≥ ρt, we can get
+d
+drG−1(r) =
+� d
+drG(r)
+�−1
+=
+t
+d
+drF(R,T )(r, t) ≤ 1
+ρ.
+(D.6)
+Then for any x ≥ 0, we have
+P
+�
+max
+0≤ℓ≤| ˆ
+Sc(t)|−1
+�
+R
+ˆ
+Sc(t)
+(ℓ+1) − R
+ˆ
+Sc(t)
+(ℓ+1)
+�
+≥ x
+ρ
+��� ˆSc(t) = Sc
+�
+(i)
+= P
+�
+max
+0≤ℓ≤|Sc|−1
+�
+RSc
+(ℓ+1) − RSc
+(ℓ)
+�
+≥ x
+ρ
+���
+�
+i∈Sc
+{Ti ≤ t}
+�
+(ii)
+= P
+�
+max
+0≤ℓ≤|Sc|−1
+�
+G−1(U(ℓ+1)) − G−1(U(ℓ))
+�
+≥ x
+ρ
+�
+(iii)
+≤ P
+�
+max
+0≤ℓ≤|Sc|−1
+�
+U(ℓ+1) − U(ℓ)
+�
+≥ x
+�
+,
+(D.7)
+where (i) holds due to (D.4) and the fact that {Ti}i∈Sc are independent of {Ti}i∈C\Sc, (ii) follows from (D.5),
+and (iii) comes from (D.6). Invoking Lemma A.4, we can finish the proof.
+D.4
+Proof of Lemma A.6
+Proof of Lemma A.6. For any absolute constant C ≥ 1, we divide the subsets of C into two groups:
+Ξ1 =
+�
+Sc ⊆ C : |Sc| > C log(n ∨ m) ·
+t2
+t2 − t1
+�
+,
+Ξ2 =
+�
+Sc ⊆ C : |Sc| ≤ C log(n ∨ m) ·
+t2
+t2 − t1
+�
+.
+Then for any x, y ≥ 0, it holds that
+P
+�
+�
+1
+| ˆSc(t2)|
+������
+�
+i∈ ˆ
+Sc(t2)
+Zi
+������
+≥ x + y
+�
+� ≤ P
+�
+�
+1
+| ˆSc(t2)|
+������
+�
+i∈ ˆ
+Sc(t2)
+Zi
+������
+≥ x1
+�
+ˆSc(t2) ∈ Ξ1
+�
++ y1
+�
+ˆSc(t2) ∈ Ξ2
+�
+�
+�
+=
+�
+Sc⊆C
+P
+�
+�
+1
+| ˆSc(t2)|
+������
+�
+i∈ ˆ
+Sc(t2)
+Zi
+������
+≥ x1
+�
+ˆSc(t2) ∈ Ξ1
+�
++ y1
+�
+ˆSc(t2) ∈ Ξ2
+� �� ˆSc(t2) = Sc
+�
+� × P
+�
+ˆSc(t2) = Sc
+�
+=
+�
+Sc∈Ξ1
+P
+�
+1
+|Sc|
+�����
+�
+i∈Sc
+1 {t1 < Ti ≤ t2}
+����� ≥ x
+�� ˆSc(t2) = Sc
+�
+P
+�
+ˆSc(t2) = Sc
+�
++
+�
+Sc∈Ξ2
+P
+�
+1
+|Sc|
+�����
+�
+i∈Sc
+1 {t1 < Ti ≤ t2}
+����� ≥ y
+�� ˆSc(t2) = Sc
+�
+P
+�
+ˆSc(t2) = Sc
+�
+.
+(D.8)
+51
+
+According to the definition of ˆSc(t2), we have
+�
+ˆSc(t2) = Sc
+�
+=
+�
+�
+�
+�
+l∈Sc
+{Tl ≤ t2} ,
+�
+l∈C\Sc
+{Tl > t2}
+�
+�
+� .
+It implies that for any i ∈ Sc,
+P
+�
+t1 < Ti ≤ t2
+��� ˆSc(t2) = Sc
+�
+= P
+�
+t1 < Ti ≤ t2
+��Ti ≤ t2
+�
+= P (t1 < Ti ≤ t2)
+P (Ti ≤ t2)
+= t2 − t1
+t2
+,
+where the first equality follows from the independence of the samples in C. Recall the definition Zi =
+1 {t1 < Ti ≤ t2} − t2−t1
+t2
+, then for any i ∈ Sc we have
+E
+�
+Zi
+��� ˆSc(t2) = Sc
+�
+= E
+�
+�Zi
+���
+�
+l∈Sc
+{Tl ≤ t2} ,
+�
+l∈C\Sc
+{Tl > t2}
+�
+�
+= P
+�
+t1 < Ti ≤ t2
+��� ˆSc(t2) = Sc
+�
+− t2 − t1
+t2
+= 0.
+(D.9)
+Next we will bound the conditional moment generating function of �
+i∈Sc Zi. For any λ > 0, we have
+E
+�
+eλ �
+i∈Sc Zi
+��� ˆSc(t2) = Sc
+�
+= E
+�
+� �
+i∈Sc
+eλZi
+���
+�
+l∈Sc
+{Tl ≤ t2} ,
+�
+l∈C\Sc
+{Tl > t2}
+�
+�
+= E
+�
+E
+� �
+i∈Sc
+eλZi
+���Ti ≤ t2, {Tl}l∈Sc\{i}
+� ���
+�
+l∈Sc
+{Tl ≤ t2}
+�
+= E
+�
+�
+�
+l∈Sc\{i}
+eλZlE
+�
+eλZi
+���Ti ≤ t2
+� ���
+�
+l∈Sc
+{Tl ≤ t2}
+�
+�
+(i)
+≤
+�
+1 + λ2E
+�
+Z2
+i eλ|Zi|���Ti ≤ t2
+��
+E
+�
+�
+�
+l∈Sc\{i}
+eλZl
+���
+�
+l∈Sc
+{Tl ≤ t2}
+�
+�
+(ii)
+=
+�
+1 + λ2E
+�
+Z2
+i eλ|Zi|���Ti ≤ t2
+��
+E
+�
+�
+�
+l∈Sc\{i}
+eλZl
+���
+�
+l∈Sc\{i}
+{Tl ≤ t2}
+�
+�
+≤
+�
+i∈Sc
+�
+1 + λ2E
+�
+Z2
+i eλ|Zi|���Ti ≤ t2
+��
+(iii)
+≤ exp
+�
+λ2 �
+i∈Sc
+E
+�
+Z2
+i eλ|Zi|���Ti ≤ t2
+��
+,
+(D.10)
+where (i) holds since (D.9) and the basic inequality ey − 1 − y ≤ y2e|y|, (ii) holds due to the independence,
+and (iii) comes from the basic inequality 1 + y ≤ ey. Notice that
+�
+i∈Sc
+E
+�
+Z2
+i eλ|Zi|���Ti ≤ t2
+�
+= |Sc|E
+�
+Z2
+1eλ|Z1|���T1 ≥ t2
+�
+≤ eλ|Sc|E
+�
+1 {t1 < Ti ≤ t2}
+���Ti ≤ t2
+�
+= eλ|Sc|t2 − t1
+t2
+=: K2
+Sc(λ),
+(D.11)
+52
+
+where the inequality holds since |Z1| ≤ 1 {t1 < Ti ≤ t2} ≤ 1. Using Markov’s inequality and (D.10), for any
+z ≥ 0, we can get
+P
+�
+��
+i∈Sc
+Zi ≥ 2KSc(1)z
+���
+�
+l∈Sc
+{Tl ≤ t2} ,
+�
+l∈C\Sc
+{Tl > t2}
+�
+�
+= P
+�
+�eλ �
+i∈Sc Zi ≥ e2λKSc(1)z���
+�
+l∈Sc
+{Tl ≤ t2} ,
+�
+l∈C\Sc
+{Tl > t2}
+�
+�
+≤ e−2λKSc(1)zE
+�
+�eλ �
+i∈Sc Zi
+���
+�
+l∈Sc
+{Tl ≤ t2} ,
+�
+l∈C\Sc
+{Tl > t2}
+�
+�
+(i)
+≤ e−2λKSc(1)z exp
+�
+λ2 �
+i∈Sc
+E
+�
+Z2
+i eλ|Zi|���Ti ≤ t2
+��
+(ii)
+≤ e−2λKSc(1)z+λ2K2
+Sc(λ),
+where (i) comes from (D.10), and (ii) comes from the definition of KSc(λ) in (D.11). By the definition of Ξ1,
+we know that KSc(1)2 ≥ C log(n ∨ m) for any Sc ∈ Ξ1. Taking z = (C log(n ∨ m))1/2, λ =
+z
+KSc(1) ≤ 1, then
+we have
+P
+������
+�
+i∈Sc
+Zi
+����� ≥ 2KSc(1)z
+��� ˆSc(t2) = Sc
+�
+≤ 2 exp
+�
+−2z2 + z2 K2
+Sc(λ)
+K2
+Sc(1)
+�
+≤ 2e−z2 = 2(n ∨ m)−C.
+(D.12)
+For Sc ∈ Ξ2 such that KSc(1)2 < C log(n ∨ m), applying (D.11) with λ = 1 gives
+P
+������
+�
+i∈Sc
+Zi
+����� ≥ 2eC log(n ∨ m)
+��� ˆSc(t2) = Sc
+�
+≤ 2e−2eC log(n∨m)E
+�
+e
+�
+i∈Sc Zi
+��� ˆSc(t2) = Sc
+�
+≤ 2e−2eC log(n∨m)+K2
+Sc(1)
+≤ 2e−2eC log(n∨m)+eC log(n∨m)
+≤ 2(n ∨ m)−C.
+(D.13)
+Taking x = 2KSc(1)(C log(n ∨ m))1/2 and y = 2eC log(n ∨ m), then plugging (D.12) and (D.13) into (D.8)
+gives
+P
+�
+�
+1
+| ˆSc(t2)|
+������
+�
+i∈ ˆ
+Sc(t2)
+Zi
+������
+≥ 2
+�
+eC log(n ∨ m)
+| ˆSc(t2)|
+�
+t2 − t1
+t2
++ 2eC log(n ∨ m)
+| ˆSc(t2)|
+�
+�
+≤ 2(n ∨ m)−C
+�
+� �
+C⊆Ξ1
+P
+�
+ˆSc(t2) = Sc
+�
++
+�
+C⊆Ξ2
+P
+�
+ˆSc(t2) = Sc
+�
+�
+�
+= 2(n ∨ m)−C.
+Thus we have complete the proof.
+53
+
+D.5
+Proof of Lemma A.7
+Proof. By the definition of ˆSc(t), we have
+| ˆSc(t)| − nt =
+�
+i∈C
+1 {Ti ≤ t} − nt =
+�
+i∈C
+1 {Ti ≤ t} − P (Ti ≤ t) .
+Applying Hoeffding’s inequality, we have
+P
+����| ˆSc(t)| − nt
+��� ≥ 2C
+�
+n log(n ∨ m)
+�
+≤ (n ∨ m)−C.
+Using the assumption 8C log(n ∨ m)/(nt) ≤ 1, we can finish the proof.
+D.6
+Proof of Lemma A.8
+Proof. Invoking the spacing representation in Lemma D.1, we have
+P
+�
+TU\{j}
+(⌈γm⌉) ≤ γ
+2
+�
+= P
+��⌈γm⌉
+i=1
+Vi
+�m
+k=1 Vi
+≤ γ
+2
+�
+= P
+�
+�
+1
+⌈γm⌉
+�⌈γm⌉
+i=1
+Vi
+1
+m
+�m
+k=1 Vk
+≤ γ
+2
+m
+⌈γm⌉
+�
+�
+≤ P
+�
+�
+1
+⌈γm⌉
+�⌈γm⌉
+i=1
+Vi
+1
+m
+�m
+k=1 Vk
+≤ 1
+2
+�
+�
+≤ P
+�
+�
+1
+⌈γm⌉
+⌈γm⌉
+�
+i=1
+Vi − 1 ≤ −1
+4
+�
+� + P
+�
+1
+m
+m
+�
+k=1
+(Vk − 1) ≥ 1
+2
+�
+≤ (n ∨ m)−C.
+54
+