From the Huang–Kotz FGM distribution to Baker’s bivariate distribution

Abstract

a b s t r a c t Huang and Kotz (1999) [17] considered a modification of the Farlie-Gumbel-Morgenstern (FGM) distribution, introducing additional parameters, and paved the way for many research papers on modifications of FGM distributions allowing high correlation. The first part of the present paper is a review of recent developments on bivariate Huang-Kotz FGM distributions and their extensions. In the second part a class of new bivariate distributions based on Baker's system of bivariate distributions is considered. It is shown that for a model of a given order, this class of distributions with fixed marginals which are based on pairs of order statistics constructed from the bivariate sample observations of dependent random variables allows higher correlation than Baker's system. It also follows that under certain conditions determined by Lin and Huang (2010) [21], the correlation for these systems converges to the maximum Fréchet-Hoeffding upper bound as the sample size tends to infinity. just its marginal distributions, since there are many admissible numbers α allowing (1) to be a bivariate distribution function. Cambanis [8] showed that for nondegenerate distributions F X and F Y , an admissible α lies between α min = − min{(

Reliability and mean residual life functions of coherent systems in an active redundancy

“…is a PQD, then bivariate order statistics (X r:n , X s:n ), 1 ≤ r < s ≤ n is also a PQD, and if (X, Y ) is an NQD, then (X r:n , X s:n ), 1 ≤ r < s ≤ n is an NQD [7,19]. Using this property, we have the following essentially from the line above (12) and (16):…”

“…It is known that if is a PQD, then bivariate order statistics is also a PQD, and if is an NQD, then is an NQD . Using this property, we have the following essentially from the line above (12) and (16):THEOREM If is PQD, then for all i.e.…”

“…The cdf of T k:n s , which is the lifetime of an (n − k + 1)-outof-n ARSL system involves the joint distribution of bivariate order statistics (X r:n , Y s:n) , 1 ≤ r < s ≤ n. For some recent results on bivariate order statistics and their connection with bivariate binomial distributions see [6,7,19,23]. …”

“…The cdf of , which is the lifetime of an ‐out‐of‐ ARSL system involves the joint distribution of bivariate order statistics For some recent results on bivariate order statistics and their connection with bivariate binomial distributions see .THEOREM The distribution function of the lifetime of the ‐out‐of‐ ARSL system is where …”

Concomitants of Order Statistics and Record Values from Generalization of FGM Bivariate-Generalized Exponential Distribution

“…is a genuine bivariate df. However, when 0 < p < 1 the model (4) becomes very poor and is not allowing any improvement of the positive correlation compared to the classical FGM model (see, [6]). Therefore most of authors who tackle the model (4) postulate the condition p > 1.…”