A More Flexible Counterpart of a Huang–Kotz's Copula-type

Abstract

We propose a more flexible symmetric counterpart of the Huang–Kotz’s copula of the 1st type. Both the counterpart and Huang–Kotz’s copula of the 1st type provide the same improvement in the correlation level. Moreover, the proposed copula includes special cases of many other extensions of the Farlie–Gumbel–Morgenstern (FGM) copula.

Scrutiny of a More Flexible Counterpart of Huang–Kotz FGM’s Distributions in the Perspective of Some Information Measures

“…where U i ∼ GE(θ 1 , (i + 1)α 1 ) and V i ∼ GE(θ 2 , (i + 1)α 2 ), for i = 1, 2. Thus, by (8), we obtain…”

“…The admissible range derived in [10] and the claim about the correlation was demonstrated to be false. The corrected admissible range of this copula was obtained in [8] and it was shown to be regarded as HK-FGM1's counterpart in that they both have two shape parameters and offer an equal improvement over the FGM copula in terms of the positive correlation between the dependent variables. The suggested copula, as an extension of the classical FGM copula, has a simpler functional form than HK-FGM1, because to obtain it, we used an additional shape parameter as a multiplicative factor, whereas to obtain the latter, the additional shape parameter is used as an exponent.…”

“…The cumulative distribution function (CDF) and probability density function (PDF) of the new extended FGM family (denoted by HK-FGM3 or more specifically HK-FGM3(a, b)) are given by (cf. [8])…”

“…and f Y (y) are the PDFs of the RVs X and Y, respectively, whereas F X and F Y are the survival functions (SFs). When the two marginals F X (x) and F Y (y) are continuous, Barakat et al [8] showed that the natural parameter space Ω (the admissible set of the parameters a and b that makes F X,Y (x, y) is a bonafide CDF) is convex, and the set Ω is given by Ω = Ω + ∪ Ω − , where…”

“…Finally, concluding remarks are discussed in Section 6 to call attention to the main innovation that sets this work apart from previous ones. To be clear, the only research that has been published is on the family HK-FGM3(a, b) by Barakat et al [8,9], which merely established the admissible range and correlation coefficient of this promising family. As a result, every discovery related to COSs and information measures is unprecedented.…”