Some properties and inferential problems related to non- normal distributions with Kurtosis Three

Abstract

Normal distribution is one of the most celebrated statistical distribution in the literature. The importance of normal distribution is mostly because of central limit theorem, one of the fundamental theorem that form a bridge between Probability and Statistics and because of its extensive applications in various scientific investigations. A large number of alternative distributions were suggested in place of normal probability densit function. For example, when the data seems to be more heavy tailed than normal, Laplace distribution can be used instead of normal. When the information is available in the form of first few moments we use normality assumption when the skewness ¯1 = 0 and kurtosis ¯2 = 3. Kale and Sebastian (1996) showed that there exist a wide class of symmetric distributions with Pearsons measure of kurtosis ¯2 = 3. A member of this class can be obtained by considering a mixture of two symmetric non-normal densities, with centers of symmetry being the same, say zero, the kurtosis of one component strictly less than 3 and that of the other component strictly greater than 3. Motivated from the examples of Kale and Sebastian (1996) in this thesis we try to identify and characterize a fairly large class of symmetric mesokurtic distributions and try to replace this class in place of normal model. We, then study the properties of the estimators, when the random sample is from one of the members of this family, in which normal distribution is a particular case. The thesis is divided into five chapters. Chapter 1 provides an introduction and summary of the thesis. In Chapter 2, we prove few characterization results of symmetric mesokurtic family. While all non-normal mesokurtic densities considered by Kale and Sebastian (1996) and others were mixtures of two symmetric probability density functions, we give an example of a class of mesokurtic distribution which cannot be considered as a mixture of any two densities. This class contains both unimodal as well as multimodal densities.