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A stochastic Expectation–Maximisation (EM) algorithm for construction of mortality tables

Published online by Cambridge University Press:  04 May 2017

Luz Judith R. Esparza  and
Fernando Baltazar-Larios
Luz Judith R. Esparza*
Affiliation:
Facultad de Ciencias Exactas, Universidad Juárez del Estado de Durango, Av. Veterinaria 210, Valle del Sur, 34120 Durango, Dgo., México
Fernando Baltazar-Larios
Affiliation:
Facultad de Ciencias, Universidad Nacional Autónoma de México, A.P. 20-726, 01000 CDMX, México
*
*Correspondence to: Luz Judith R. Esparza, Facultad de Ciencias Exactas, Universidad Juárez del Estado de Durango, Av. Veterinaria 210, Valle del Sur, 34120 Durango, Dgo., México. Tel. (+52) 5951040830; E-mail: judithr19@gmail.com
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Abstract

In this paper, we present an extension of the model proposed by Lin & Liu that uses the concept of physiological age to model the ageing process by using phase-type distributions to calculate the probability of death. We propose a finite-state Markov jump process to model the hypothetical ageing process in which it is possible the transition rates between non-consecutive physiological ages. Since the Markov process has only a single absorbing state, the death time follows a phase-type distribution. Thus, to build a mortality table the challenge is to estimate this matrix based on the records of the ageing process. Considering the nature of the data, we consider two cases: having continuous time information of the ageing process, and the more interesting and realistic case, having reports of the process just in determined times. If the ageing process is only observed at discrete time points we have a missing data problem, thus, we use a stochastic Expectation–Maximisation (SEM) algorithm to find the maximum likelihood estimator of the intensity matrix. And in order to do that, we build Markov bridges which are sampled using the Bisection method. The theory is illustrated by a simulation study and used to fit real data.

Keywords

MortalityPhysiological agePhase-type distributionsEM algorithmBisection
Type
Papers
Information
Annals of Actuarial Science , Volume 12 , Issue 1 , March 2018 , pp. 1 - 22
Copyright
© Institute and Faculty of Actuaries 2017 

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References

Aalen, O.O. (1995). On phase type distributions in survival analysis. Scandinavian Journal of Statistics, 22, 447–463.Google Scholar
Asmussen, S. & Hobolth, A. (2012). Markov bridges, bisection and variance reduction. In L. Plaskota & H. Wozniakowski, (Eds.), Monte Carlo and Quasi-Monte Carlo Methods 2010 (Springer Proceedings in Mathematics and Statistics, Volume 23, (pp. 3–22). Springer, Berlin, Heidelberg.Google Scholar
Billingsley, P. (1961). Statistical Inference for Markov Processes. University of Chicago Press, Chicago, IL.Google Scholar
Bladt, M. (2005). A review on phase–type distributions and their use in risk theory. Astin Bulletin, 35(1), 145–161.Google Scholar
Bladt, M., Esparza, L.J.R. & Friis, B.F. (2011). Fisher information and statistical inference for phase-type distributions. Journal of Applied Probability, 48A, 277–293.Google Scholar
Bladt, M. & Sorensen, M. (2005). Statistical inference for discretely observed Markov jump processes. Journal of the Royal Statistical Society, 67, 395–410.Google Scholar
Bobbio, A., Horvath, A., Scarpa, M. & Telek, M. (2003). Acyclic discrete phase type distributions: properties and a parameter estimation algorithm. Performance Evaluation: An International Journal, 54, 1–32.Google Scholar
Bobbio, A. & Telek, M. (1994). A benchmark of Ph estimation algorithms: results for acyclic-PH. Stochastic Models, 10, 661–677.Google Scholar
Booth, H. & Tickle, L. (2008). Mortality modelling and forecasting: a review of methods. Annals of Actuarial Science, 3, 3–43.Google Scholar
Breuer, L. & Kume, A. (2010). An EM algorithm for Markovian arrival processes observed at discrete times. LNCS, 5987, 242–258.Google Scholar
Celeux, G. & Diebolt, J. (1986). The SEM algorithm: a probabilistic teacher algorithm derived from the EM algorithm for mixture problem. Computational Statistics Quarterly, 2, 599–613.Google Scholar
Craik, A. (2002). Edward Sang (1805–1890): calculator extraordinary’, special number of Newsletter in memory of J.G. Fauvel. British Society for the History of Mathematics Newsletter, 45, 32–45.Google Scholar
Davies, E.B. (2010). Emdebbdable markov matrices. Electronic Journal of Probability, 15(47), 1474–1486.Google Scholar
Dempster, A.P., Rubin, D.B. & Laird, N.M. (1977). Maximum likelihood from incomplete data via the EM algorithm (with discussion). Journal of the Royal Statistical Society B, 39, 1–38.Google Scholar
Gompertz, B. (1825). On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. Philosophical Transactions of the Royal Society of London, 115, 513–583.Google Scholar
Halley, E. (1693). An estimate of the degrees of the mortality of mankind, drawn from curious tables of the births and funerals at the city of Breslau; with an attempt to ascertain the price of annuities upon lives. Philosophical Transactions of the Royal Society of London, 17, 596–610.Google Scholar
Heligman, M.A.L. & Pollard, J.H. (1980). The age pattern of mortality. Journal of the Institute of Actuaries, 107, 49–80.Google Scholar
Hobolth, A. & Jensen, J.L. (2011). Summary statistics for end-point conditioned continuous-time Markov chains. Journal of Applied Probability, 48, 911–924.Google Scholar
Hobolth, A. & Stone, E. (2009). Efficient simulation from finite-state, continuous-time Markov chains with incomplete observations. Annals of Applied Statistics, 3, 1204–1231.Google Scholar
Hobolth, A. & Thorne, J.T. (2014). Sampling and summary statistics of endpoint-conditioned paths in DNA sequence evolution. In M.H. Chen, L. Kuo & P.O. Lewis (Eds.), Bayesian Phylogenetics: Methods, Algorithms, and Applications. Chapman and Hall/CRC.Google Scholar
Jacobsen, M. (1982). Statistical analysis of counting processes. Lecture Notes in Statistics, 12.Google Scholar
Jones, H.B. (1956). A special consideration of the aging process, disease and life expectancy. In J.H. Lawrence & C.A. Tobias, (Eds.), Advances in Biological and Medical Physics, Volume 4, (pp. 281–337).Google Scholar
Knowles, D.A., Part, S.L., Glass, D. & Winn, J.M. (2011). Inferring a measure of physiological age from multiple ageing related phenotypes. NIPS Workshop From Statistical Genetics to Predictive Models in Personalized Medicine. Sierra Nevada, Spain. December 2011.Google Scholar
Küchler, H. & Sorensen, M. (1997). Exponential Families of Stochastic Processes. Springer, New York.Google Scholar
Latouche, G. & Ramaswami, V. (1999). Introduction to Matrix Analytic Methods in Stochastic Modeling. Society for Industrial and Applied Mathematics.CrossRefGoogle Scholar
Lee, R.D. & Carter, L. (1992). Modeling and forecasting the time series of U.S. mortality. Journal of the American Statistical Association, 87, 659–671.Google Scholar
Lin, X.S. & Liu, X. (2007). Markov aging process and phase-type law of mortality. North American Actuarial Journal, 11(4), 92–109.Google Scholar
Loprinzi, P.D., Branscum, A., Hanks, J. & Smit, E. (2016). Healthy lifestyle characteristics and their joint association with cardiovascular disease biomarkers in US adults. Mayo Clinic Proceedings, 9(14), 432–442.Google Scholar
Makeham, W.M. (1860). On the law of mortality and the construction of annuity tables. The Assurance Magazine, and Journal of the Institute of Actuaries, 8(06), 301–310.Google Scholar
McLachlan, G.J. & Krishnan, T. (1997). The EM Algorithm and Extensions. Wiley, New York.Google Scholar
Moivre, A.D. (1725). Annuties Upon Lives or The Valuation of Annuities Upon Any Number of Live; as alfo, of Reversions. W.P. and sold by Francis Fayram.Google Scholar
Neuts, M.F. (1975). Probability distributions of phase-type. Liber Amicorum Prof. Emeritus H. Florin, pp. 173–206.Google Scholar
Neuts, M.F. (1981). Matrix Geometric Solutions in Stochastic Models. Volume 2. Johns Hopkins University Press, Baltimore, MD.Google Scholar
Nielsen, S.F. (2000). The stochastic EM algorithm: estimation and asymptotic results. Bernoulli, 6(3), 457–489.Google Scholar
Okamura, H., Dohi, T. & Trivedi, S. (2009). Markovian arrival process parameter estimation with group data. IEEE/ACM Transactions on Networking, 17(4), 1326–1339.Google Scholar
Okamura, H., Dohi, T. & Trivedi, S. (2012). Improvement of expectation–maximization algorithm for phase-type distributions with grouped and truncated data. Applied Stochastic Models in Business and Industry, 29(2), 141–156.Google Scholar
Tataru, P. & Hobolth, A. (2012). Comparison of methods for calculating conditional expectations of sufficient statistics for continuous time Markov chains. BMC Bioinformatics, 12, 465.Google Scholar
Wei, G. & Tanner, M.A. (1990). A Monte Carlo implementation of the EM algorithm and the poor man’s data augmentation algorithm. Journal of the American Statistical Association, 85, 699–704.Google Scholar
Xu, J., Murphy, S.L., Kochanek, K.D. & Bastian, B.A. (2016). Deaths: final data for 2013. National Vital Statistics Reports: from the Centers for Disease Control and Prevention, National Center for Health Statistics, National Vital Statistics System, 64(2), 1–119.Google Scholar
Zuev, S.M., Yashin, A.L., Manton, K.G., Dowd, E., Pogojev, I.B. & Usmanov, R. (2000). Vitality index in survival modeling: how physiological aging influences mortality. Journal of Gorontology, 55A, 10–19.Google Scholar