Graduating Mortality Rates by Mixture of Pareto, Loglogistic, and Two Weibull Distributions using Bayesian Method

Christian Evan Chandra, Sarini Abdullah

Abstract


Mortality rates are important in conducting the pricing and valuation of life insurance policies. Raw values are usually wiggly to plot, and practitioners often graduate them to obtain smoothness. Current mortality models have problems related to the goodness of fit, interpretability, and usability without implementing other actuarial assumptions for fractional ages. This study proposes a mixture of Pareto, log-logistic, and two Weibull distributions with eleven parameters to graduate mortality rates. Lifespan covered are whole life, including childhood, adolescence, senescence, and the late elderly's phase. We adjusted the parameterization to improve the ease of model's interpretability right after obtaining the value of estimates. Prior distributions of the parameters and sampling model form for the data are also proposed to estimate the parameters' value using the Bayesian method with Gibbs sampling. High values of coefficient of determination produced by model fit into several data support the graphical evidence to show the model's goodness of fit and best fit occurs for the life table of Israeli males in 1987. Gelman-Rubin statistic is also very close to one and shows fast convergence in estimating the parameters. Based on the results, obtaining the best and worst estimates of newborn survival probabilities is possible. We also showed that this model could be implemented on annual and abridged mortality rates.

Keywords


Bayesian method; mixing distribution; mortality graduation; newborn survival probabilities; parametric model.

Full Text:

PDF

References


D. Dickson, M. Hardy, M. Hardy, and H. Waters, Actuarial Mathematics for Life Contingent Risks, 2nd ed. New York: Cambridge University Press, 2018.

“A practitioner’s guide to statistical mortality graduation | SOA.†https://www.soa.org/resources/tables-calcs-tools/2018-stat-mort-graduation/ (accessed Jan. 03, 2021).

H. Yamazaki, “SMT 2018 (Standard Mortality Table 2018),†Institute of Actuaries of Japan, 2017.

AAJI, “Tabel Mortalitas Indonesia IV,†Asos. Asuransi Jiwa Indones., vol. 1, no. October, 2019.

J. Marín-García, M. J. Goldenthal, and G. W. Moe, “Post-genomic view of aging: definitions, theories and observations,†in Aging and the Heart: A Post-Genomic View, Springer US, 2008, pp. 3–31.

R. Fivaz-Silbermann, Nouvelles recherches sur l’Ecole de Lausanne. Geneve, 1976.

L. Heligman and J. H. Pollard, “The age pattern of mortality,†J. Inst. Actuar., vol. 107, no. 1, pp. 49–80, 1980.

P. Dellaportas, A. F. M. Smith, and P. Stavropoulos, “Bayesian analysis of mortality data,†J. R. Stat. Soc. Ser. A Stat. Soc., vol. 164, no. 2, pp. 275–291, 2001, doi: 10.1111/1467-985X.00202.

S. Mazzuco, B. Scarpa, and L. Zanotto, “A mortality model based on a mixture distribution function,†Popul. Stud. (NY)., vol. 72, no. 2, pp. 191–200, May 2018, doi: 10.1080/00324728.2018.1439519.

M. Engelman, H. Caswell, E. A.-D. Research, and U. 2014, “Why do lifespan variability trends for the young and old diverge? A perturbation analysis,†ncbi.nlm.nih.gov, Accessed: Dec. 03, 2020. [Online]. Available: https://www.ncbi.nlm.nih.gov/pmc/articles/pmc4326020/.

M. Bebbington, C. Lai, R. Z.-J. of T. Biology, and U. 2007, “Modeling human mortality using mixtures of bathtub shaped failure distributions,†Elsevier, Accessed: Aug. 20, 2020. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0022519306005534.

J. de Beer and F. Janssen, “A new parametric model to assess delay and compression of mortality,†Popul. Health Metr., vol. 14, no. 1, Dec. 2016, doi: 10.1186/s12963-016-0113-1.

A. Feckler, M. Low, J. P. Zubrod, and M. Bundschuh, “When significance becomes insignificant: Effect sizes and their uncertainties in Bayesian and frequentist frameworks as an alternative approach when analyzing ecotoxicological data,†Environ. Toxicol. Chem., vol. 37, no. 7, pp. 1949–1955, Jul. 2018, doi: 10.1002/etc.4127.

S. C. Smid, D. McNeish, M. Miocevic, and R. van de Schoot, “Bayesian versus frequentist estimation for structural equation models in small sample contexts: A systematic review,†Structual Equ. Model. A Multidiscip. J., vol. 27, no. 1, pp. 131–161, Jan. 2020, doi: 10.1080/10705511.2019.1577140.

M. Flor, M. Weiß, T. Selhorst, C. Müller-Graf, and M. Greiner, “Comparison of Bayesian and frequentist methods for prevalence estimation under misclassification,†BMC Public Health, vol. 20, no. 1, pp. 1–10, Jul. 2020, doi: 10.1186/s12889-020-09177-4.

G. B. King, A. E. Lovell, L. Neufcourt, and F. M. Nunes, “Direct comparison between Bayesian and frequentist uncertainty quantification for nuclear reactions,†Physiscal Rev. Lett., vol. 122, no. 23, p. 232502, 2019, Accessed: Jun. 22, 2021. [Online]. Available: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.122.232502.

M. A. J. S. van Boekel, “On the pros and cons of Bayesian kinetic modeling in food science,†Trends Food Sci. Technol., vol. 99, pp. 181–193, 2020, Accessed: Jun. 22, 2021. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0924224419304170.

M. Safari, N. Masseran, K. I.-P. A. S. M. and, and undefined 2018, “Optimal threshold for Pareto tail modelling in the presence of outliers,†Phys. A Stat. Mech. Its Appl., vol. 509, pp. 169–180, 2018.

M. Raschke, “Alternative modelling and inference methods for claim size distributions,†Ann. Actuar. Sci., vol. 14, no. 1, pp. 1–19, 2020.

F. Ashkar, I. Ba, and E. Volpi, “Selection between the generalized Pareto and kappa distributions in peaks-over-threshold hydrological frequency modelling,†Hydrol. Sci. J., vol. 62, no. 7, pp. 1167–1180, May 2017, doi: 10.1080/02626667.2017.1302089.

H.-H. Kwon, D. Han, and D.-I. Kim, “Bias correction of daily precipitation over South Korea from the long-term reanalysis using a composite Gamma-Pareto distribution approach,†Hydrol. Res., vol. 50, no. 4, pp. 1138–1161, 2019, doi: 10.2166/nh.2019.127.

S. Surendran and K. Tota-maharaj, “Effectiveness of log-logistic distribution to model water-consumption data,†J. Water Supply Res. Technol., vol. 67, no. 4, pp. 375–383, 2018.

S. Markov, N. Kyurkchiev, A. Iliev, and A. Rahnev, “A note on the log-logistic and transmuted log-logistic models: some applications,†Dyn. Syst. Appl., vol. 27, no. 3, pp. 593–607, 2018, doi: 10.12732/dsa.v27i3.9.

N. Kyurkchiev and M. Svetoslav, “A family of recurrence generated sigmoidal functions based on the Log-logistic function: Some approximation aspects,†Biomath Commun., vol. 5, no. 1, 2018.

A. Arfeen, K. Pawlikowski, D. McNickle, and A. Willig, “The role of the Weibull distribution in modelling traffic in internet access and backbone core networks,†J. Netw. Comput. Appl., vol. 141, pp. 1–22, 2019.

E. E. Elmahdy, “Modelling reliability data with finite Weibull or Lognormal mixture distributions,†Appl Math Inf. Sci, vol. 11, no. 4, pp. 1081–1089, 2017, doi: 10.18576/amis/110414.

A. Chowdury, “Constitutive modelling and Weibull statistical analysis for the porosity-mechanical property correlations in 3% yittria-stabilized zirconia system,†Int. J. Refract. Met. Hard Mater., vol. 70, pp. 246–252, 2018.

S. A. Klugman, H. H. Panjer, and G. E. Willmot, Loss Models: From Data to Decisions, 5th ed. New Jersey: John Wiley & Sons, 2019.

Y.-K. Tse, Nonlife Actuarial Models: Theory, Methods and Evaluation. New York: Cambridge University Press, 2009.

B. P. Singh, S. Singh, and U. D. Das, “A general class of new continuous mixture distribution and application,†J. Math. Comput. Sci., vol. 11, no. 1, pp. 585–602, 2021, doi: 10.28919/jmcs/5227.

E. Williford, V. Haley, L. A. McNutt, and V. Lazariu, “Dealing with highly skewed hospital length of stay distributions: The use of Gamma mixture models to study delivery hospitalizations,†PLoS One, vol. 15, no. 4, Apr. 2020, doi: 10.1371/journal.pone.0231825.

A. Ickowicz and R. Sparks, “Modelling hospital length of stay using convolutive mixtures distributions,†Stat. Med., vol. 36, no. 1, pp. 122–135, 2017.

Q. F. Gronau, A. Ly, and E. J. Wagenmakers, “Informed Bayesian t-Tests,†Am. Stat., vol. 74, no. 2, pp. 137–143, Apr. 2020, doi: 10.1080/00031305.2018.1562983.

V. Fortuin, A. Garriga-Alonso, M. van der W.-S. Impacts, and undefined 2021, “Speech enhancement using Bayesian wavenet,†Interspeech, pp. 2013–2017, 2017, Accessed: Jun. 22, 2021. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S2665963821000270.

A. Gelman, D. Simpson, and M. Betancourt, “The prior can often only be understood in the context of the likelihood,†Entropy, vol. 19, p. 555, 2017, doi: 10.3390/e19100555.

J. Piironen and A. Vehtari, “On the hyperprior choice for the global shrinkage parameter in the horseshoe prior,†Artif. Intell. Stat., pp. 905–913, 2017, Accessed: Jun. 22, 2021. [Online]. Available: https://proceedings.mlr.press/v54/piironen17a.html.

M. C. Axelsen, J. R. M. Jepsen, and N. Bak, “The choice of prior in Bayesian modeling of the information sampling task,†Biol. Psychiatry, vol. 83, no. 12, p. 60, 2021, doi: 10.1016/j.biopsych.2017.04.021.

World Bank, “Mortality rate, infant (per 1,000 live births) | Data,†World Bank, 2019. https://data.worldbank.org/indicator/SP.DYN.IMRT.IN (accessed Aug. 12, 2020).

C. O. Schell, H. Rosling, S. Peterson, A. Mia Ekström, and M. Reilly, “Socioeconomic determinants of infant mortality: A worldwide study of 152 low-, middle-, and high-income countries,†Scand. J. Public Health, vol. 35, no. 3, pp. 288–297, 2007, doi: 10.1080/14034940600979171.

M. M. Williams, B. R. Kemp, K. F. Ferraro, and S. A. Mustillo, “Avoiding the major causes of death: Does childhood misfortune reduce the likelihood of being disease free in later life?,†Journals Gerontol. - Ser. B Psychol. Sci. Soc. Sci., vol. 74, no. 1, pp. 170–180, Jan. 2019, doi: 10.1093/geronb/gby039.

C. E. Chandra and S. Abdullah, “Estimating Indonesian complete life table and fair annual pure premium range from abridged life table with Bayesian method and bootstrapping.†Under review in a peer-reviewed journal, 2021.

L. Steinberg et al., “Around the world, adolescence is a time of heightened sensation seeking and immature self-regulation,†Dev. Sci., vol. 21, no. 2, p. e12532, Mar. 2018, doi: 10.1111/desc.12532.

A. Remund, C. G. Camarda, and T. Riffe, “A cause-of-death decomposition of young adult excess mortality,†Demography, vol. 55, no. 3, pp. 957–978, Jun. 2018, doi: 10.1007/s13524-018-0680-9.

S. Jafarzadeh, M. R. Golrokhian Sani, N. Mokhtari Amirmajdi, and M. Firouzi, “Evaluation of bilateral vestibular dysfunction in Iranian adults and elderlies by electronystagmography and video head impulse test,†Iran. J. Otorhinolaryngol., vol. 31, no. 2, pp. 103–1101, 2019, doi: 10.22038/ijorl.2018.32238.2062.

S. Watanabe et al., “Treatment outcomes of Burr-Hole surgery for chronic subdural hematoma in the elderly living beyond life expectancy: A study comparing cure, recurrence, and complications in patients aged ≥80 years versus ≤79 years,†World Neurosurg., vol. 132, pp. e812–e819, Dec. 2019, doi: 10.1016/j.wneu.2019.08.003.

C. W. Jordan, Society of Actuaries’ Textbook on Life Contingencies. Chicago: The Society of Actuaries, 1991.

S. Chandra, Christian Evan; Abdullah, “Forecasting mortality trend of Indonesian old aged population with Bayesian method.†Under review in a peer-reviewed journal, 2021.

United Nations, “World population prospects - population division - United Nations,†Department of Economic and Social Affairs Population Dynamics, 2020. https://population.un.org/wpp/default.aspx?aspxerrorpath=/wpp/Download/Standard/Mortality/WPP2019_MORT_F17_3_ABRIDGED_LIFE_TABLE_FEMALE.xlsx. (accessed Oct. 02, 2020).

Y. Huang, “Is centenarian rate independent from economy?,†Arch. Gerontol. Geriatr., vol. 93, p. 104312, Mar. 2021, doi: 10.1016/j.archger.2020.104312.

T. V. Pyrkov, K. Avchaciov, A. E. Tarkhov, L. I. Menshikov, A. V. Gudkov, and P. O. Fedichev, “Longitudinal analysis of blood markers reveals progressive loss of resilience and predicts human lifespan limit,†Nat. Commun., vol. 12, no. 1, p. 2765, Dec. 2021, doi: 10.1038/s41467-021-23014-1.

D. Marcotte and D. Allard, “Gibbs sampling on large lattice with GMRF,†Comput. Geosci., vol. 111, pp. 190–199, 2018.

Y. Huang, J. L. Beck, and H. Li, “Bayesian system identification based on hierarchical sparse Bayesian learning and Gibbs sampling with application to structural damage assessment,†Comput. Methods Appl. Mech. Eng., vol. 318, pp. 382–411, 2017.

Y. Huang and J. L. Beck, “Full Gibbs sampling procedure for Bayesian system identification incorporating sparse Bayesian learning with automatic relevance determination,†Comput. Civ. Infrastruct. Eng., vol. 33, no. 9, pp. 712–730, 2018.

HMD, “Sweden, Total Population, Life Tables (period 1x1), Males,†Human Mortality Database, 2020. https://www.mortality.org/cgi-bin/hmd/country.php?cntr=SWE&level=1 (accessed Oct. 20, 2020).

F. Colchero and B. Y. Kiyakoglu, “Beyond the proportional frailty model: Bayesian estimation of individual heterogeneity on mortality parameters,†Biometrical J., vol. 62, no. 1, pp. 124–135, Jan. 2020, doi: 10.1002/bimj.201800280.

HMD, “Israel, Total Population, Life Tables (period 1x1), Males,†Human Mortality Database, 2020. https://www.mortality.org/cgi-bin/hmd/country.php?cntr=ISR&level=1 (accessed Oct. 20, 2020).

HMD, “Republic of Korea, Total Population, Life Tables (period 1x1), Females,†Human Mortality Database, 2020. https://www.mortality.org/cgi-bin/hmd/country.php?cntr=KOR&level=1 (accessed Oct. 20, 2020).

N. Li, “Estimating complete life tables for populations with limited size: From graduation to equivalent construction,†Taylor Fr., vol. 24, no. 1, pp. 22–35, Jan. 2019, doi: 10.1080/10920277.2019.1585879.

L. Vu, J. Pulerwitz, B. Burnett-Zieman, C. Banura, J. Okal, and E. Yam, “Inequitable gender norms from early adolescence to young adulthood in Uganda: Tool validation and differences across age groups,†J. Adolesc. Heal., vol. 60, no. 2, pp. S15–S21, Feb. 2017, doi: 10.1016/j.jadohealth.2016.09.027.

K. L. Humphreys et al., “Foster care promotes adaptive functioning in early adolescence among children who experienced severe, early deprivation,†J. Child Psychol. Psychiatry Allied Discip., vol. 59, no. 7, pp. 811–821, Jul. 2018, doi: 10.1111/jcpp.12865.

Y. Hori, A. Matsumura, T. Namikawa, M. Kato, M. Iwamae, and H. Nakamura, “Comparative study of the spinopelvic alignment in patients with idiopathic lumbar scoliosis between adulthood and adolescence,†World Neurosurg., vol. 149, pp. e309–e315, May 2021, doi: 10.1016/j.wneu.2021.02.031.

S. Krivenko, V. Lukin, O. Krylova, and V. Shutko, “Visually lossless compression of retina images,†in 2018 IEEE 38th International Conference on Electronics and Nanotechnology, ELNANO 2018 - Proceedings, Sep. 2018, pp. 255–260, doi: 10.1109/ELNANO.2018.8477459.




DOI: http://dx.doi.org/10.18517/ijaseit.12.5.15220

Refbacks

  • There are currently no refbacks.


Published by INSIGHT - Indonesian Society for Knowledge and Human Development