The Farlie-Gumbel-Morgenstern copula approach has always been of interest to
researchers as a tool for introducing bivariate models. However, this approach
is not suitable to deal with the data that are highly dependent. It provides a
better description of the data whose maximum correlation coefficient is 0:333:
This degree of correlation occurs when the uniform marginal distribution is used
in the Farlie-Gumbel-Morgenstern copula method. However, the marginal uniform
distribution is not always applicable. By searching the statistical literature,
we observe that different margin distributions such as exponential, generalized
exponential, gamma, and Rayleigh etc., have been considered in the
Farlie-Gumbel-Morgenstern bivariate distribution family and have studied their
correlation. So far in the literature, researchers have adopted only the classical/traditional distributions
using the Farlie-Gumbel-Morgenstern approach. In this thesis, we consider
a family of distributions as a marginal distribution in the Farlie-Gumbel-
Morgenstern approach. This is one of the key core of this thesis as it can cover all
the specific distributions. The second goal of this thesis is to carry out the statistical
properties using a comprehensive structure of generalized order statistics.
In this thesis, we first consider the extended Weibull family as a marginal distribution
in the Farlie-Gumbel-Morgenstern method. Then we obtain some basic
properties and formulas for calculating the correlation coefficient. Furthermore,for a specific case of the proposed family, the degree of correlation is calculated
mathematically and numerically via a simulation study. Moreover, the general
results for measures of entropy and entropy of the generalized order statistics
are obtained. Based on the ranked set sampling, the Bayesian estimators are also
obtained.
Next, we consider and study a new extropy measure based on generalized order
statistics. In this way, numerous extropy measures such as extrop