Bayesian Estimation for {ij}-Inflated Mixture Power Series Distributions using an EM Algorithm

Amir  T. Payandeh Najafabadi; Mansoureh  Sakizadeh

Authors

Amir T. Payandeh Najafabadi Department of Mathematical Sciences, Shahid Beheshti University, G.C. Evin, Tehran, Iran
Mansoureh Sakizadeh Department of Mathematical Sciences, Shahid Beheshti University, G.C. Evin, Tehran, Iran

Keywords:

Power series distribution/regression, mixture model, inflated model, EM algorithm, rate-making system

Abstract

The purpose of this article was to illustrate how we can model different behaviors of subpopulations by introducing a mixing distribution/regression model with {ij}-inflated power series. An EM algorithm was used to estimate the parameters of the models. As a practical application, the new model has been applied to the design of an optimal rate-making system. More precisely, this article
employs a number of reported claims from an Iranian third party insurance dataset, under an {ij} inflated power series mixture distribution/regression model to estimate rate premium for such insurance contract. The numerical study shows that the {ij}-inflated negative Binomial mixture models have the potential to provide more appropriate rate premiums for policyholders under a rate-making system with four categories.

References

Alshkaki RSA. On the zero-one inflated poisson distribution. J Stat Distrib Appl. 2016; 2(4): 42-48.

Aryuyuen S, Bodhisuwan W. The negative binomial-generalized exponential (NB-GE) distribution.

Appl Math Sci. 2013; 7(22): 1093-105.

Bermudez L, Karlis D. Bayesian multivariate poisson models for insurance ratemaking. Insur Math ´

Econ. 2011; 48(2), 226-36.

Boucher JP, Inoussa R. A posteriori ratemaking with panel data. Austin Bull. 2014; 44(3): 587-612.

Denuit M, Dhaene J. Bonus-Malus scales using exponential loss functions. Bl atter der DGVFM.

; 25(1): 13-27.

Denuit M, Marechal X, Pitrebois S, Walhin JF. Actuarial modelling of claim counts: Risk classifica- ´

tion, credibility and bonus-malus systems. New York: John Wiley & Sons; 2007.

Dionne G, Vanasse C. A generalization of automobile insurance rating models: the negative binomial

distribution with a regression component. Austin Bull. 1989; 19(2): 199-212.

Dionne G, Vanasse C. Automobile insurance ratemaking in the presence of asymmetrical information.

J Appl Econ. 1992; 7(2): 149-165.

Greene WH. Accounting for excess zeros and sample selection in poisson and negative binomial

regression models. Technical Report [No. EC-94-10]. Department of Economics, Stern School

of Business: New York University.

Gupta PL, Gupta RC, Tripathi RC. Analysis of zero-adjusted count data. Comput Stat Data Anal.

; 23(2): 207-218.

Hall DB. Zero-inflated poisson and binomial regression with random effects: a case study. J Biom.

; 56(4): 1030-1039.

Heras A, Moreno I, Vilar-Zanon JL. An application of two-stage quantile regression to insurance ´

ratemaking. Scand Actuar J. 2018; 9: 1-17.

Johnson NL, Kemp AW Kotz S. Univariate Discrete Distributions (Vol. 444). New York: John Wiley

& Sons; 2007.

Joshi R. A Generalized Inflated Geometric Distribution. PhD [dissertation]. Marshal university; 2015.

Khalid M, Aslam M, Sindhu TN. Bayesian analysis of 3-components Kumaraswamy mixture model:

quadrature method vs. Importance sampling. Alexandria Eng J. 2020; 59(4): 2753-2763.

Lange JT. Application of a mathematical concept of risk to property-liability insurance ratemaking. J

Risk Insur. 1969; 38(2): 383-391.

Lee GY, Frees EW. General Insurance Deductible Ratemaking with Applications to the Local Government Property Insurance Fund. Available from:

https://www.soa.org/Files/Research/Projects/research-2016-gi-deductible-ratemaking.pdf.

Lemaire, J. (1995). Bonus-malus systems in automobile insurance. New York: Springer science &

business media; 1995.

Lim HK, Li WK, and Philip LH. Zero-inflated poisson regression mixture model. Comput Stat Data

Anal. 2014; 71: 151-158.

Liu W, Tang Y, and Xu A. Zero-and-one-inflated poisson regression model. Stat Pap (Berl). 2021;

(2):, 915-934.

McLachlan G, and Krishnan N. The EM algorithm and Extensions. New York: Wiley; 1997.

Melkersson M, and Rooth D. Modeling female fertility using inflated count data models. J Popul

Econ. 2000; 13: 189-203.

Nishii R, and Tanaka S. Modeling and inference of forest coverage ratio using zero-one inflated

distributions with spatial dependence. Environ Ecol Stat. 2013; 20(2): 315-336.

Payandeh Najafabadi, AT. A new approach to the credibility formula. Insur Math Econ. 2010; 46(2):

-338.

Payandeh Najafabadi, AT, Atatalab F, and Omidi Najafabadi M. Credibility premium for rate-making

systems. Commun Stat Theory Methods. 2017; 46(1): 415-426.

Payandeh Najafabadi AT, and Sakizadeh M. Designing an Optimal Bonus–Malus System Using the

Number of Reported Claims, Steady-State Distribution, and Mixture Claim Size Distribution.

Int J Ind Syst Eng. 2019; 32(3): 304-331.

Pinquet, J. Allowance for cost of claims in bonus-malus systems. Austin Bull. 1997; 27(1): 33-57

Pinquet, J. (1998). Designing optimal bonus-malus systems from different types of claims. Austin

Bull. 1998; 28(2): 205-220.

Razie F, Bahrami Samani E, and Ganjali M. A latent variable model for analyzing mixed longitudinal

{k, l}-inflated count and ordinal responses. J Appl Stat. 2016; 43(12), 2203-2224.

Rigby RA, and Stasinopoulos DM. The GAMLSS project: a flexible approach to statistical modelling. In New trends in statistical modelling: Proceedings of the 16th international workshop

on statistical modelling. University of Southern Denmark; 2001, p. 345-355.

Simar, L. Maximum likelihood estimation of a compound poisson process. Ann Stat. 1976; 4(6):

-1209.

Simon, LJ. Fitting negative binomial distributions by the method of maximum likelihood. Proc. Casual. Actuar. Soc. 1961; 48: 45-53.

Sindhu TN, and Aslam M. Preference of prior for Bayesian analysis of the mixed Burr type X distribution under type I censored samples. Pakistan J Stat Operat Res. 2014; 10(1): 17-39.

Sindhu TN, Aslam M, and Hussain Z. A simulation study of parameters for the censored shifted

Gompertz mixture distribution: A Bayesian approach. J stat Manag Syst. 2016; 19(3): 423-450.

Sindhu TN, Hussain Z, and Aslam M. On the Bayesian analysis of censored mixture of two ToppLeone distribution. Sri Lankan J Appl Stat. 2019; 19(1): 13-20.

Sindhu TN, Riaz M, Aslam M, and Ahmed Z. Bayes estimation of Gumbel mixture models with

industrial applications. Trans Ins Meas Control. 2016; 38(2): 201-214.

Sindhu TN, Hussain Z, and Aslam M. Parameter and reliability estimation of inverted Maxwell mixture model. J stat Manag Syst. 2019; 22(3): 459-493.

Sindhu TN, Khan HM, Hussain Z, and Al-Zahrani, B. Bayesian inference from the mixture of halfnormal distributions under censoring. J Natl Sci Found. 2018; 46(4): 587-600.

Tzougas G, and Frangos N. The design of an optimal bonus-malus system based on the Sichel distribution. Mod Probl Insur Math. 2014; 1: 239-260.

Tzougas G, Vrontos S, and Frangos N. Optimal bonus-malus systems using finite mixture models.Austin Bull. 2014; 44(2): 417-444.

Walhin JF, and Paris J. Using Mixed poisson Processes in Connection with Bonus-Malus Systems.

Austin Bull. 1999; 29(1): 81-99.

Wieczorek J, Nugent C, and Hawala S. A Bayesian zero-one inflated beta model for small area shrinkage estimation. In Proceedings of the 2012 Joint Statistical Meetings, American Statistical Association, Alexandria, VA; 2012.