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

Christian Evan Chandra, Sarini Abdullah


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.


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

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