Some generalizations of cauchy distribution

Abstract

This thesis is mainly concerned with study of some generalizations of Cauchy distribution. The distributions commonly used for modelling of insurance losses, financial returns, file sizes on the network servers, etc. are subject to some sort of deficiencies. Also, there are only few probability distributions capable of modelling heavy tailed data sets and none of them are flexible enough to provide greater accuracy in fitting complex forms of data. Furthermore, in financial and actuarial risk management problems, the data sets are usually unimodal, skewed to the right, and possess thick right tail. The distributions that exhibit such characteristics can be used quite effectively to model insurance loss data to estimate the business risk level. To address the problems stated above, we have an interest in defining new families of distributions through different approaches such as introducing additional, location, scale, shape, and transmuted parameters, to generalize the existing distributions. We adopted six estimation approaches for estimating parameters of our models, and assess the performance of these estimators. Therefore, this study is addressed scientific computation challenge by performing numerical comparison between several estimators for the model parameters and identified which of them perform better in terms of estimation efficiency. The comparison is based on Monte Carlo simulations and the outcomes of a real data analysis. The simplicity of the proposed distributions and the great flexibility in modelling real life data will attract researchers to use these distributions as an alternative of the Cauchy distribution in modelling different scenarios. Our proposed families are useful for modelling insurance claim data sets, better models for financial returns because the normal model does not capture the large fluctuations seen in real assets. Also, our families of distributions have received considerable attention due to the heavy tail property.