A study on discrete distributions and integer valued autoregressive INAR processes

Abstract

The main objectives of the present research work are concerned with the study on some discrete distributions and integer-valued autoregressive processes. It also concentrates on studying various generalizations of discrete distributions like discrete Mittag-Leffler, discrete stable-Linnik, geometric discrete semi stable-Linnik, Lu¨ ders Formel I, Delaporte, discrete Poisson-Laplace, Katz Family of distributions, etc. Characteristic properties of the new models are investigated, the advantages of these models over the base models are established and finally various applications of the newly developed models are explored. The thesis consists of 7 Chapters. Chapter 1 serves as an introduction, which gives a survey of literature relating to the subject matter of the present study, the basic concepts and notations used in the thesis and finally a summary of the work executed as part of the study. Generalization of discrete Mittag-Leffler distribution is introduced and studied in Chapter 2. Chapter 3 deals with the integer valued autoregressive processes with a convolution of discrete stable and discrete Linnik distributions and their generalization as marginals. Chapter 4 introduces a new stationary integer valued time series model with a special form of the negative binomial marginal distribution, which has received much attention during recent years. We obtain some properties of the distribution and estimate the moments of the innovation processes. Chapter 5 concentrates on the Delaporte distribution. Chapter 6 reviews various entropy measures in information theory. The importance of discrete Laplace distribution and the key concept of information theory namely, Shan- non entropy and other generalizations are discussed. Chapter 7 proposes a new class of stationary first order integer-valued autoregressive processes with Katz family of marginal distributions using the binomial thinning operator newline