Estimating a Common Parameter of Two Farlie-Gumbel-Morgenstern Bivariate Uniform Distributions by Induced Ranked Set Sampling

Ranked set sampling from several populations

In this paper, we introduce a new generalization of univariate and bivariate modified discrete Weibull distribution. Various properties of the univariate generalized modified discrete Weibull distribution, such as survival function, probability mass function, hazard rate function, probability generating function, and moment generating function, are derived. The joint distribution function, joint probability mass function, marginal distributions, moment generating function, and conditional distribution of the proposed bivariate distribution are derived. Parameters of the distributions are estimated using maximum likelihood estimation. The use of these distributions is illustrated using real-life datasets.

The method of ranked set sampling when units are to be induced from several bivariate populations is introduced in this work. The best linear unbiased estimator (BLUE) of the common parameter θ₂, based on n ranked set observations, is obtained when n₁ observations are drawn from one of the two Farlie-Gumbel-Morgenstern (FGM) bivariate uniform distributions and n₂ observations, drawn from the other FGM bivariate uniform distribution such that n = n₁ + n₂. The BLUE of θ₂ based on the upper extreme ranked set sample is also obtained. These estimators are compared with the usual BLUE of the common parameter θ₂ based on the concomitants of order statistics of an equivalent sample size.