Abstract
Kenyan insurance firms have introduced insurance policies of chronic illnesses like cancer; however, they have faced a huge challenge in the pricing of these policies as cancer can transit into different stages, which consequently leads to variation in the cost of treatment. This has made the estimation of aggregate losses of diseases which have multiple stages of transitions such as cancer, an area of interest of many insurance firms. Mixture phase type distributions can be used to solve this setback as they can in-cooperate the transition in the estimation of claim frequency while also in-cooperating the heterogeneity aspect of claim data. In this paper, we estimate the aggregate losses of secondary cancer cases in Kenya using mixture phase type Poisson Lindley distributions. Phase type (PH) distributions for one and two parameter Poisson Lindley are developed as well their compound distributions. The matrix parameters of the PH distributions are estimated using continuous Chapman Kolmogorov equations as the disease process of cancer is continuous while severity is modeled using Pareto, Generalized Pareto and Weibull distributions. This study shows that aggregate losses for Kenyan data are best estimated using PH-OPPL-Weibull model in the case of PH-OPPL distribution models and PH-TPPL-Generalized Pareto model in the case of PH-TPPL distribution models. Comparing the two best models, PH-OPPL-Weibull model provided the best fit for secondary cancer cases in Kenya. This model is also recommended for different diseases which are dynamic in nature like cancer.
Highlights
Aggregate losses are estimated by in-cooperating both claim frequency and claim severity distributions
This study shows that aggregate losses for Kenyan data are best estimated using Phase type (PH)-OPPL-Weibull model in the case of PH-OPPL distribution models and PH-TPPL-Generalized Pareto model in the case of PH-TPPL distribution models
Unlike ordinary distributions which do not in-cooperate the transition of different states, the distributions proposed here take into consideration transition states while modeling claim frequency data
Summary
Aggregate losses are estimated by in-cooperating both claim frequency and claim severity distributions. Transform in estimation of aggregate losses from frequency and severity distributions. Mohamed et al (2010) [4] introduced use of simulation approach in estimation of aggregate losses which can be employed when frequency and severity distribution cannot be combined to derive a compound distribution. Aggregate loss distributions are based on collective risk model expressed as: N. i =1 where: Xi is the severity distribution and N is the claim count distribution. Wu et al (2010) [10] developed phase type distributions when frequency distributions followed Panjer class (a,b, 0) while Kok et al (2010) [11] used phase type distributions of Panjer class (a,b,1) to model claim frequency. The concept of Markov chains is used to determine the matrices of the phase type distributions used in modeling claim frequency. Poisson distribution is often used to model count data; it is based on the assumption that variance to mean ratio is unity (equi-dispersion) which is not applicable to real data; it is considered as
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