Estimation of parameters of certain thick-tailed distributions with applications

Abstract

In recent years the theory and application of thick-tailed distributions has received considerable attention in literature, especially in economic framework. The stable distributions, Students-t distribution, and mixture of normals have been used to describe the behavior of stock price changes. The Pareto distribution has been used to model fire losses, incomes, etc. In this thesis, the estimation of the parameters of (i) stable distributions, (ii) Students-t distribution, (iii) mixture of normals, and (iv) Pareto distribution are investigated. Three methods of estimation, namely, (i) the c.f. method (Paulson et. al. (1975)), (ii) maximum likelihood method, and (iii) minimum Cramer-von Mises method are used to estimate parameters. A constrained estimation technique is developed to obtain better fits in the tail. The c.f. method is extended to the estimation of parameters of symmetric and asymmetric Cauchy and three parameter gamma.

Two insurance claim processes are investigated in the framework of stable laws and Pareto distribution. The stable law, Students-t and the mixture of normals are investigated in the framework of stock price changes.