diff --git "a/3tAzT4oBgHgl3EQfuf3H/content/tmp_files/load_file.txt" "b/3tAzT4oBgHgl3EQfuf3H/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/3tAzT4oBgHgl3EQfuf3H/content/tmp_files/load_file.txt" @@ -0,0 +1,780 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf,len=779 +page_content='Mortality modeling at old-age: an mixture model approach Silvio C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Patricio* The Interdisciplinary Centre on Population Dynamics, University of Southern Denmark Fredy Castellares Departamento de Estat´ıstica, Universidade Federal de Minas Gerais Bernardo Queiroz Departamento de Demografia, Universidade Federal de Minas Gerais January 5, 2023 Abstract In this paper, we propose a mixture-based model for mortality modeling above age 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' The proposed model is compared with 4 other widely used models: the Beard, Gompertz, Makeham, and Perks models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Our model captures well the mortality rate’s behavior at all the ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' We applied the method to a country with high quality data, Japan, and one with lower data quality, Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' In the comparative study for the Japanese population, the model presented a better fit to the data, obtaining an absolute mean percentage error of less than 7%, while the other models presented values greater than 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Keywords: mixture model, old-age, mortality modeling 1 Introduction In the past centuries, much has been done to model the process of mortality in populations and its con- sequences (Graunt, 1662;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Gompertz, 1825a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Wilmoth, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' van Raalte, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' One of humanity’s most outstanding achievements in the last century, perhaps the last millennium, has been the four-decade increase in human life expectancy over the past 160 years (Vaupel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Wilmoth, 2000) and the improvement in human mortality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' All these changes in human longevity directly affect pension, welfare, and health care systems (Cutler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' silca@sam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='sdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='dk 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='01693v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='AP] 4 Jan 2023 Despite pioneering work by Graunt and Gompertz, understanding of mortality for older ages remains a challenge, specially in developing countries with more defective data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' In general, mortality estimates at older ages are limited by small numbers both in the exposure, death count and problems with age declaration (Feehan, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Wrigley-Field, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Nepomuceno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' There is an important and ongoing debate about the levels of mortality at older ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' In general terms, the debate is whether mortality at older ages is declining or continues to increase (Gavrilov & Gavrilova, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Feehan, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' In some settings, such as Brazil, there is also an important question on the crossover of mortality at older ages when comparing different population sub-groups (Nepomuceno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Pinheiro & Queiroz, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Gomes & Turra, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' In addition to the problem of the quality of the data, there is a debate on hypotheses of selectivity and of the biological limit of mortality in human populations that, in different ways, would impact the behavior of mortality taxes in more advanced ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' One of the consequences of the mortality selectivity hypothesis would be a greater rate of deceleration of the rates of mortality in more advanced ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' In this context, there are a series of models to explain mortality behavior at older ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' The choice of the appropriate model depends on the hypotheses assumed, whether in relation to the quality of the two data or in relation to the impacts produced by the selectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' There are several possible explanations for the observed results and estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' First one is related to data quality in different areas of a country, across sub-population groups and age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' For instance, it could be a consequence of different age misreporting patterns or issues with quality of vital registration systems (Black et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Preston et al (2000) investigated how different types of age misreporting can affect estimates of mortality rates at older ages, by analyzing the effects of three patterns of age misreporting: net age overstatement, net age understatement, and symmetric age misreporting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='. It is also possible that mortality selection plays a role in the observed levels of mortality at older ages (Barbi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Wachter, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' In the context of higher mortality rates at young ages, survivors to older ages would be physiologically stronger and then live longer than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Unfortunately, data quality at older ages limits the understanding of mortality and the evolution of survivorship at older ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Feehan (2018) uses alternative methods to cohort mortality above age 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' He finds that no model can be universally applied to estimate old-age mortality, but he argues that Log-Quad (Wilmoth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=', 2012) provides a good fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' However, the log-quad method is based on standard mortality changes from the Human Mortality Database that is constructed from a series of countries in the Northern Hemisphere and might be limited to low and middle income countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' In this paper, we suggest a model that captures decline in mortality rates at older ages, which is a characteristic observed in some populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Based on the proposed model, we perform a comparative study using establish mortality laws with our proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' The analysis was split into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' First, to 2 compare the four widely used models with the proposed model: in this part we will study the behavior of these models in two databases: one with good quality data on mortality in Japan in 2015 (obtained from The Human Mortality Database of mortality), and the other database that has limited data regarding mortality in Brazil in 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' In it the models will be evaluated from Mean Absolute Percentage Error (MAPE) of the log-hazard using the leave-one-out cross-validation method, and the model with the least MAPE will all be the best model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Moreover, as some models are complex, the genetic algorithm was used to obtain the estimates via maximum likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Using this algorithm ensures convergence to the global maximum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' The second part applies the proposed model to different databases, and aims to understand the model behavior and also to verify its potential for application to real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='The model presented a better fit to the data, obtaining an absolute mean percentage error of less than 7%, while the other models presented values greater than 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' 2 Models specification’s and parameter estimation Considering a non negative random variable (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=') T defined in a probability space (R+, B, Pθ), representing the individual life-spam, the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' T can be characterized by the survival function S(x|θ) = Pθ(T > x) which is associated with the density f(x|θ) = − ∂ ∂xS(x|θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' If S is a continuous survival function associated with a f density function, then the function µ defined in R+ by µ(x|θ) = lim ε↓0 Pθ(x < T < x + ε|X > x) ε = f(x|θ) S(x|θ) it’s called the T mortality force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' This function is usually used to describe the force of mortality for a group of people or population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' The inferences in the model are based on the assumption that the number of death has a Poisson dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Therefore, be D = (D0, D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' , Dm)′ a random sample with Poisson distribution, with Dk representing the number of deaths between ages [k, k + 1), with k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' , m, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' the number of death of people with k years old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' For this approach it is considered that E(Dk) = µ(k|θ)Ek, with µ(k|θ) representing the mortality force at age k, where θ = (θ1, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' , θp)′ is the parameter vector that characterizes the mortality rate, and Ek the population at age k exposed to risk, that are assumptions widely used by demographers (Brillinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=', 3 1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Also, as it is the Poisson distribution, we have to V(Dk) = µ(k|θ)Ek, same value of expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Be D = (D0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' , Dm)′ e E = (E0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' , Em)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' The log-likelihood function from θ is given by ℓ(θ|D) = m � k=1 Dk log λ(θ, k) − λ(θ, k), (1) with λ(θ, x) = µ(x|θ)E(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' The likelihood estimate �θ is obtained from maximizing the log-likelihood function with in equation 1, with respect to θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Obtaining the partial derivative vector of the equation 1, with respect to θi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' , p, we have ∂ℓ(θ|D) ∂θi = m � k=1 � Dk µ(k|θ) − Ek � ∂µ(k|θ) ∂θi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' (2) The likelihood estimation can also be obtained by equating the partial derivative vector to zero and simul- taneously solving the system of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' The explicit form of the gradient vector is explained for each of the models considered in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' The Newton-Raphson method can be applied to solve the likelihood equation to obtain the estimate �θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='1 Beard model In this model introduced in Beard (1959), we have that the force of mortality is given by µ(k|θ) = aebk 1 + δebk with θ = (a, b, δ)′ ∈ R3 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' From which we calculate the partial derivative with respect to a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' E Equation 2 gives us a general equation for the gradient vector, where it depends only on the mortality rate and its partial derivative with respect to each parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Hence we get ∂ℓ(θ|D) ∂a = m � k=1 � Dk �1 + δebk aebk � − Ek � ebk (1 + δebk) ∂ℓ(θ|D) ∂b = m � k=1 � Dk �1 + δebk aebk � − Ek � akebk (1 + δebk)2 ∂ℓ(θ|D) ∂δ = m � k=1 � Dk �1 + δebk aebk � − Ek � ae2bk (1 + δebk)2 4 representing the gradient vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='2 Gompertz model In this model introduced in Gompertz (1825b), we have that the force of mortality is given by µ(k|θ) = aebk, with θ = (a, b)′ ∈ R2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' So for the gradient vector we have ∂ℓ(θ|D) ∂a = m � k=1 � Dk aebk − Ek � ebk ∂ℓ(θ|D) ∂b = m � k=1 � Dk aebk − Ek � akebk 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='3 Makeham model In this model introduced in Makeham (1860), we have that the force of mortality is given by µ(k|θ) = aebk + c, with θ = (a, b, c)′ ∈ R3 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' So for the gradient vector we have ∂ℓ(θ|D) ∂a = m � k=1 � Dk aebk + c − Ek � ebk ∂ℓ(θ|D) ∂b = m � k=1 � Dk aebk + c − Ek � akebk ∂ℓ(θ|D) ∂c = m � k=1 � Dk aebk + c − Ek � 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='4 Perks model In this model introduced in Perks (1932), we have that the force of mortality is given by µ(k|θ) = γ + aebk 1 + δebk with θ = (a, b, γ, δ)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' So for the gradient vector we have ∂ℓ(θ|D) ∂a = m � k=1 � Dk � 1 + δebk γ + aebk � − Ek � ebk 1 + δebk ∂ℓ(θ|D) ∂b = m � k=1 � Dk � 1 + δebk γ + aebk � − Ek � k(a − δγ)ebk (1 + δebk)2 ∂ℓ(θ|D) ∂γ = m � k=1 � Dk � 1 + δebk γ + aebk � − Ek � 1 1 + δebk ∂ℓ(θ|D) ∂δ = m � k=1 � Dk � 1 + δebk γ + aebk � − Ek � ebk � aebk + γ � (1 + δebk)2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='5 Mixture model As with Makeham, we will seek to decompose mortality into two components: premature and senescent mortality, respectively modeled by an exponential and a Gompertz component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' However, Makeham dis- tinguishes these components through mortality force, and here we propose to distinguish them through distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Therefore, we are considering that the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' T introduced at the beginning of this session is associated with a probability density function f, which is define as: f(x|θ) = p � λe−λx� + (1 − p) � ab exp � a � ebx − 1 � + bx �� (3) with θ = (a, b, λ, p)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' The density f is a Gompertz and a exponential distribution a mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' The Gompertz distribution will fit the senescence deaths count, and the exponential distribution will fit the premature deaths, such as accidents and disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Briefly, this model considers the existence of two sub populations in the death count, one 6 Gompertz and the other Exponential, and the parameters p and q = 1 − p represent the proportions of each one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Since the random variable T is associated with a density function, we can also associate it with a hazard function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' In this case the force of mortality is defined by: µ(x|θ) = f(x|θ) S(x|θ) = p � λe−λx� + (1 − p) � ab exp � a � ebx − 1 � + bx �� pe−λx + (1 − p) exp{−a (ebx − 1)} , (4) for which there is no straightforward interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Which is lost due to the ease of deriving functions such as statistical moments and expected average residual life (for more details, see Finkelstein (2009)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='From 7 this we can get the gradient vector which,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' for this model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' is given by ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='∂ℓ(θ|D) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='∂a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='p (λe−λk) + (1 − p) (ab exp {a (ebk − 1) + bk}) − Ek ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='× b(1 − p)ea(keb−1)+bλ + ab(1 − p)(keb − 1)ea(keb−1)+bx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='pe−λk + (1 − p) exp{−a (ebk − 1)} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='+ (−1)(1 − p)(1 − ebk)e−a(ebk−1) � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='λe−λk� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='+ (1 − p) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='(pe−λk + (1 − p) exp{−a (ebk − 1)})2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='∂ℓ(θ|D) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='∂b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='p (λe−λk) + (1 − p) (ab exp {a (ebk − 1) + bk}) − Ek ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='a(1 − p)xebx−a(ebx−1) � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='ab(1 − p)ea(xebx−1)+bx + λpe−λx� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='(1 − p)ea(ebx−1) + pe−λx�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='+ a(1 − p)ea(ebx−1)+bx + ab(1 − p)ea(ebx−1)+bx � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='axeb + x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='(1 − p)ea(ebx−1) + pe−λx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='∂ℓ(θ|D) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='∂λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='k=1 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='p (λe−λk) + (1 − p) (ab exp {a (ebk − 1) + bk}) − Ek ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='pe−λx − λpxe−λx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='(1 − p)e−a(ebx−1) + pe−λx + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='pxe−λx � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='ab(1 − p)ea(xeb−1)+bx + λpe−λx� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='(1 − p)e−a(ebx−1) + pe−λx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='∂ℓ(θ|D) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='∂p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='k=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='Dk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='pe−λk + (1 − p) exp{−a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='ebk − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='p (λe−λk) + (1 − p) (ab exp {a (ebk − 1) + bk}) − Ek ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='�� λe−λx − abea(xeb−1)+bx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='(1 − p)e−a(ebx−1) + pe−λx − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='e−λx − e−a(ebx−1)� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='ab(1 − p)ea(xeb−1)+bx + λpe−λx� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='(1 − p)e−a(ebx−1) + pe−λx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='Data and empirical results ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='In order to evaluate the proposed model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' we will compare its performance on high and low-quality data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' For this, we will evaluate its performance against four other models, using the Mean Absolute Percentage Error (MAPE) combined with the leave-one-out cross-validation method, which will measure the average distance between the log-hazard and the log-mortality rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Moreover, as some models are highly nonlinear, 8 the Genetic Algorithm (Scrucca, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Mirjalili, 2019) will be used to maximize the likelihood function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' This algorithm ensures convergence to the global maximum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='1 Models comparison In a high quality data setting In this scenario, we will use mortality data from Japan in 2015 obtained from The Human Mortality Database (HMD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' The observed value of log µ is linearly increasing to a certain age, and then has a sharp drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' This behavior was also noted this country in the last three decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' However this is not restricted to Japan, other countries like Sweden, Germany, USA and Korea also had the same mortality behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' The Figure 1 shows the estimated log-hazard function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' We can clearly see the models of Beard, Gompertz, Makeham and Perks were not able to fit properly the mortality rate after age 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' 70 80 90 100 110 −5 −4 −3 −2 −1 0 1 Age log(µ) Beard Gompertz Makeham Perks Mixture model Figure 1: Japan 2015 modeling The Gompertz model consider force of mortality being log-linear, but clearly this behavior does not describe the entire observed curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' For this model the estimated parameter is �θ = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='0179, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='1094)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' And this model has a MAPE of 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='0127, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=', this model’s predictions are on average 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='0127 % distant of the observed value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' A similar result can be obtained from the Makeham model, which has estimated parameter �θ = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='0174, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='1103, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='0008)′, and MAPE 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='0288.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' The Beard can be seen as the ratio of a Gompertz and a Makeham models with c = 1, with the pa- rameters estimated by ML �θ = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='0165, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='1216, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='0073)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Despite Beard’s combination of Makeham and Gompertz models, this model provided the worse fit, reaching a MAPE of 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='6189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' The Perks model also has a similar construction to Beard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' It is the ratio between two Makeham models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' For this model we estimate �θ = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='0135, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='1313, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='0040, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='0075)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' And as expected, this model had a very 9 similar behavior to the previous model, including in MAPE of 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='3591, suggesting that this model does not fit well to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Finally, for the proposed mixture-based model, we estimated �θ = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='1155, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='0163, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='2061, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='0126)′, and a MAPE of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='9193, the best of the models presented in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' In addition, this model was the only one that was able to capture the sharp drop in the mortality rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' With the estimated parameters we can interpret that the non-senescence death represents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='2599 % of the total death after age 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' In a low quality data setting We observed that the model works well on data that has good quality, and now we aim to understand how the model behaves when the data has limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' In this case we are going to use data from Brazil from 2010 (Queiroz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Gonzaga & Schmertmann, 2016) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Previous studies showa a mortality crossover above age 60 when comparing more and less developed states in Brazil using the Topals model (Queiroz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Gonzaga & Schmertmann, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' It is argued that the result is related to the level of complete- ness of death counts, age misreporting and mortality selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Thus, it is an important and relevant case study for the application of our prosed mixture model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' For this, as before, we will compare the performance of the 5 models presented through MAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' 70 75 80 85 90 95 100 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='0 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='5 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='0 Age log(µ) Beard Gompertz Makeham Perks Mixture model Figure 2: Brazil 2010 modeling For the first model (Beard) we estimated �θ = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='0375, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='0942, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='5625 × 10−8)′, and a MAP of 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='4629, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=', on average this model distanced by 20 % of the mortality rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' We also got a similar conclusion about the Gompertz model, estimating �θ = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='0375, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='0943)′ and MAPE about 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='4499.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' The Makeham and Perks models also obtained similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' For Makeham it was estimated �θ = 10 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='01481, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='1338, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='03131)′ resulting in a MAP of 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='5473, and for Perks model it was estimated �θ = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='0163, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='0129, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='0290, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='4272 × 10−7)′ which results in MAPE of 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='9002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Finally, for the proposed model was estimated �θ = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='1036, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='0315, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='2389, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='0692)′, and a MAPE of 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='0038%, which indicates that the model is not able to capture mortality well in these data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Therefore, the results found in this application match the results discussed in Feehan (2018) on the power of models capturing mortality at advanced ages universally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='2 Model applications As we have seen, the proposed model has a high capacity to fit the mortality at older ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Therefore, we will illustrate the power of this model by applying it to mortality data from Japan (1993 and 2002), Sweden (2011), Germany (2016), USA (1990 and 1992), Spain (2012) and Italy (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Table 1 represents the estimate for each dataset, and Figure 4 represents their respective decomposed distribution of death.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Table 1: Parameters estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Country Year ˆa ˆb ˆc ˆp MAPE Japan 1993 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='10911 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='02916 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='21615 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='00250 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='86459 Japan 2002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='10897 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='02425 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='30152 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='03276 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='49451 Sweden 2011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='12390 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='01520 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='26448 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='01559 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='27019 Germany 2016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='11046 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='02090 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='22283 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='00397 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='68258 USA 1990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='08845 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='03569 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='20360 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='02569 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='80694 USA 1992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='09057 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='03404 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='20575 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='03217 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='91887 Spain 2012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='12372 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='01544 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='22751 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='01307 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='38755 Italy 2011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='11606 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='01768 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='21710 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='01999 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='24385 In Table 1 it can be seen that the estimated values for p are small, less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='04, which indicates that the proportion of premature deaths above age 70 does not exceed 4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' This result was already expected, since by truncating the mortality data at age 70, we are excluding infant mortality and mortality hump (Remund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=', 2018), and we only observe the tail of the distribution of premature mortality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Furthermore, the our result is also in agreement with Horiuchi & Wilmoth’s results, that above age 75 mortality decelerates for most causes of death (Horiuchi & Wilmoth, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' The estimated values for the c parameter are similar, and concentrated around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' This suggests that, despite having different proportions, the distributions of premature death are similar, as can be seen on the left in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Such similarity was not observed in the senescent death distributions, which have a marked difference, as can be seen on the right in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Despite this, it is clear that the modal age of death is between 80 and 90, which is consistent with previous studies and (Horiuchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' 11 70 80 90 100 110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='30 Age prematur mortality distribution JPN − 1993 JPN − 2002 SWE − 2011 DEUTNP − 2016 USA − 1990 USA − 1992 ESP − 2012 ITA − 2011 70 80 90 100 110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='05 Age senescence mortality distribution JPN − 1993 JPN − 2002 SWE − 2011 DEUTNP − 2016 USA − 1990 USA − 1992 ESP − 2012 ITA − 2011 Figure 3: Estimates of mortality components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='110 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='10000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='20000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='30000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='JPN − 1993 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='Age ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='110 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='10000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='20000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='30000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='JPN − 2002 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='Age ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='110 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='SWE − 2011 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='Age ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='110 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='10000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='20000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='30000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='DEUTNP − 2016 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='Age ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='110 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='20000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='40000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='60000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='USA − 1990 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='Age ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='110 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='20000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='40000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='60000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='USA − 1992 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='Age ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='110 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='10000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='15000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='ESP − 2012 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='Age ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='110 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='15000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='25000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='ITA − 2011 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='Age ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='Senescence deaths ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='Premature deaths ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='Overall deaths ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='Figure 4: Estimations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='The Figure 4 shows the distribution of death estimated and broken down into premature and senescent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='deaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' In it, we can observe the quality of fit of the estimated model (black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' In addition, it is possible 12 to see that for Japan in 1993 and Germany in 2016, there were practically no premature deaths after age 70, this could also be inferred from analyzing the Table 1, where the values of estimate for p are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' 4 Conclusions and future works Robust estimates of mortality rates in advanced ages are a challenge for demographers for various reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Even in populations with good records of deaths and population there are disturbances in the function of the low number of events and/or some limitation in the information on the age of death.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' In case of countries where the problems of data quality is present, the challenges are greater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' For some centuries there has been an ambition to decompose mortality into interpretable components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' The best known are those proposed by Makeham (1860) and Heligman & Pollard (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' However, in recent years researchers have devoted to this problem (Remund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Mazzuco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Therefore, this paper aims to bring a contribution to this discussion, delivering a new parametric model capable of decomposing mortality through mixing models in a frequentest framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Mazzuco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' (2021) proposes an approach similar to the one proposed in this paper, however the authors use a Bayesian framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' As we have seen, the proposed model fits well the mortality curve, specially above age 100, and this model does it without overparametrization, as Heligman & Pollard (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Furthermore, as it is a mixture model, the model is flexible to become the Gompertz model (p = 0), or the Exponential model (p = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' When 0 < p < 1, the model fits a mortality curve with inflexion point (mortality deceleration) and plateau (mortality plateau).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' The use of Brazilian mortality data shed light on the performance of the model in a low quality database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' We could see that the mixture-based model captures the dynamics of mortality well only when there is a drop in mortality rates, serving as an alternative to models that do not have this characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Although the present work presents a model capable of capturing the specific dynamics of the force of mortality in certain populations, it also sheds light on other problems to be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Since the model is based on mixtures of distributions, we are interested in deriving hypothesis tests on the estimated parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' One of the main ones is to test if p = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' whether the model can be reduced to a Gompertz model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' similar interest to that studied in B¨ohnstedt & Gampe (2019), when a hypothesis test for Gamma heterogeneity is derived, and important statistical properties are studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' Finally, in the recently published paper Vaupel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' (2022) point out that estimating senescence mor- tality is of fundamental importance to understand the pace of human aging, human longevity and how far we can live.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} +page_content=' In this sense, this work brought a method capable of identifying and estimating senescent mor- tality, without having a great computational cost, often seen in Bayesian analysis (See Barber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf'} 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