A STUDY ON CONCOMITANTS OF ORDERED VARIABLES

Abstract

Concomitants of ordered variables have recently shown their importance in many fields such as biological, physical, industrial and economical disciplines. Also, one of the most important bivariate distributions, due to its practical applications, is the Farlie-Gumbel-Morgenstern (FGM) family of distributions.

The aim of this thesis is to study concomitants of ordered variables arising from some generalized forms of the FGM family of distributions.

The thesis consists of five chapters.

Chapter 1 Introduction and Basic Concepts

Chapter 1 presents a brief literature review concerning the point of the research. The definitions and the basic concepts that are used throughout the thesis are also introduced.

Chapter 2 Bivariate Farlie-Gumbel-Morgenstern Distributions

In Chapter 2, we discuss the classic FGM family of bivariate distributions. We also discuss some extensions of the classic FGM family of bivariate distributions that proposed by Huang-Kotz (1999) and by Bairamov, Kotz and Bekci (2001), by introducing additional parameters in order to increase the correlation between the variables of the distribution in the case of uniform marginals. We introduce general forms for these extensions for any continuous marginal distributions. We study the correlation structure of these general extensions with respect to the additional parameters for the cases of the exponential marginal distributions and the logistic marginal distributions.

The new results of this chapter are published in both "Journal of Advances in Mathematics", U.S.A, 7(3). 2014. 1328-1340, and "Journal of Statistical Science and Application", 2, 2014. 175-192,U.S.A

Chapter 3 Concomitants of Record Values Arising from Generalized Farlie-Gumbel-Morgenstern Distributions

Chapter 3 deals with the distributions of concomitants of record values arising from the generalized forms proposed in Chapter 2. We derive the probability density function of the concomitant of the nth record value and the joint probability density function of the concomitants of the mth and nth, m<n, record values. The kth moment of the concomitant of the nth upper record and the product moment of concomitants of the mth and nth upper record values for m<n are also presented. Then the results obtained are investigated for some important members of the family namely, two-parameter exponential marginal distributions and two-parameter logistic marginal distributions. For both cases, best linear unbiased estimators (BLUEs), based on concomitants of record values, of some parameters involved in the distribution are derived. A rough moment type estimator of the association parameter is obtained. Two predictors of future concomitants of record values are presented, a best linear unbiased predictor (BLUP) and a conditional predictor. The BLUP is obtained using the well known BLUP method, while we have proposed the second predictor, which we call conditional predictor, depending on conditional distribution. Finally numerical illustrations are presented to highlight the theoretical results obtained.