Generalized Poisson-Lindley Distribution for Overdispersion in Insurance Claims Data
Utriweni Mukhaiyar(a*), Hamidah Qurrotun Nadwah(b), Arli Maghfirah Utami(b), Elisa Murti Dewi(b), Rizka Puspitasari(b)

a) Statistics Research Division, Faculty of Mathematics and Natural Science,
Institut Teknologi Bandung, Jl. Ganesha No. 10, Bandung 40132, Indonesia
*utriweni.mukhaiyar[at]itb.ac.id
b) Master Program in Actuarial Science, Faculty of Mathematics and Natural Science,
Institut Teknologi Bandung, Jl. Ganesha No. 10, Bandung 40132, Indonesia

Abstract

A count data describes events within a certain period of time. The values of count data are only positive integers. Therefore, count data capable to describe the occurrence of insurance claims in a company. This study uses data on health insurance claims from the Health Insurance Administration for West Java region in 2016 as data to be tested for the type of distribution. Commonly, Poisson distribution can be used to describe this kind of data. Meanwhile, these data is known that the value of zero has an excessive
frequency compared to other values. It is also known that variance value of data exceeds the average value, which is called overdispersion. Therefore, Poisson distribution can not be used in this data, as it violates Poisson distribution requisite that the mean and variance are same. Here, we compare Negative Binomial distribution and Generalized Poisson-Lindley distribution to describe the data. Generalized Poisson-Lindley distribution is an extended version of compound Poisson distribution which is obtained by mixing Poisson distribution with Generalized Poisson-Lindley distribution. Parameter values of Generalized Poisson-Lindley distribution can be estimated using maximum likelihood estimation method through Newton Raphson iteration numerical method. According to research results, it was determined that Generalized Poisson-Lindley distribution was appropriate to describe the distribution of overdispersion data which is compared with other distribution, specifically Negative Binomial distribution and Poisson distribution. This is evidenced by a comparison of the Akaike Information Criterion values to other distributions for overdispersion cases. In addition, results show that tolerance limit for overdispersion data when using Generalized Poisson-Lindley distribution is 0.222 to 0.513.

Keywords: overdispersion, generalized Poisson-Lindley, AIC criteria, overdispersion tolerance limit

Topic: MATHEMATICS AND STATISTICS