Phase Type Models Applied in Estimation of Aggregate Claim Losses of Secondary Cancer Cases

Abstract

Aggregate losses can be applied widely in areas of actuarial science as well as financial mathematics. They can be calculated using the collective risk model which sums random losses involving both claim severity and claim frequency. Impact of claim severity on aggregate losses has been well explored in previous research while less research has been done on impact of claim frequency on aggregate losses especially using phase type distributions which motivates this study. In this research we improve on calculation of aggregate losses by introducing phase type distributions in modeling claim frequency, construct phase type Poisson Lindley, determine their properties and parameter estimation. This research also determines how to get matrix parameters of phase type distributions, construct phase type compound probability generating function and apply the proposed models to secondary cancer cases in Kenya to demonstrate their advantage. Phase type distributions have one of their parameter as a matrix hence they can be used to model claim frequency for diseases which have multiple stages of transition and data which applies bonus malus system. The phase type distributions considered in this research are Panjer class (a, b, 0) , class (a, b, 1) and Poisson Lindley distributions. Matrices calculated using Chapman-Kolmogorov equation have shown to fit well in the phase type distributions. The concept of survival analysis (Kaplan-Meier) is used to estimate the transition probabilities of the matrix parameters and the long run probabilities represent the row vector →Y. Severity distributions considered are one and two parameter Poisson Lindley distribution, Pareto, Generalized Pareto and Wei-bull distributions. Method of moments is used in estimation of parameters of the severity distributions while Panjer recursive model and Discrete Fourier Transform are used in estimation of aggregate loss probabilities. Phase type distributions, help us investigate the impact of frequency within frequency in estimation of aggregate losses. PH Poisson-Generalized Pareto model provided the best fit for Panjer class (a, b, 0) while PH ZT Poisson-Generalized Pareto model provided the best fit for class (a, b, 1) and PH two parameter Poisson Lindley-Generalized Pareto model provided the best fit for Poisson Lindley distributions. Finally, we propose phase type two parameter Poisson Lindley-Generalized Pareto as the best overall model for modeling secondary cancer data in Kenya and similar data. This research enables the insurance sector to improve its reserving models for cancer which has become a world wide menace.