Correction to the paper “A new extension of the FGM copula with an application in reliability” by Rasha Ebaid, Walid Elbadawy, Essam Ahmed & Abdalla Abdelghaly

Ebaid et al. (2020) proposed the following new extended FGM copula (1) $$C(u,v)=uv[1+a(1-u)(1-v)(1-bu)(1-bv)], 0\le b<1.$$(1) Ebaid et al. (2020) claimed that, the admissible range of the parameters is (2) $$-1\le a\le \frac{1}{1-b} \text{and} 0\le b<1.$$(2) Based on the admissible range (2)(2) $$-1\le a\le \frac{1}{1-b} \text{and} 0\le b<1.$$(2) , Ebaid et al. (2020) claimed that the Spearman’s rho, $\rho \in (-0.333,1),$ and Kendall’s tau, $\tau \in (-0.222,1).$ Unfortunately, these ranges of the correlations, as well as the given admissible range (2)(2) $$-1\le a\le \frac{1}{1-b} \text{and} 0\le b<1.$$(2) , are wrong as the following example shows:

Counter example

1. Pick out two admissible values b = 0.9 and $a=9.99,$ clearly $\rho >1,$ which doesn’t make sense?

2. Pick out the admissible value $b=0.9.$ It can be shown that the probability density function of the copula (1)(1) $$C(u,v)=uv[1+a(1-u)(1-v)(1-bu)(1-bv)], 0\le b<1.$$(1) at $u=\frac{2}{3}$ and v = 0 is negative for all admissible values $a\in (3,10].$

Remark.

Now the important question is: Does the copula (1)(1) $$C(u,v)=uv[1+a(1-u)(1-v)(1-bu)(1-bv)], 0\le b<1.$$(1) (of course with the true admissible range which was not determined for the two parameters a and b in Ebaid et al., 2020, or even in this correction) increase the maximum positive correlations for the FGM copula $C_0(u,v)=uv[1+\alpha (1-u)(1-v)], -1\le \alpha \le 1?$ Unfortunately, the answer is no, as the following example shows: Consider the FGM family with the two marginals $G_1(u)=u[1+b(1-u)]$ and $G_2(v)=v[1+b(1-v)],0\le u,v\le 1, 0\le b<1.$ (3) $$C^{\ast }(u,v)=uv[1+b(1-u)][1+b(1-v)][1+a(1-u)(1-v)(1-bu)(1-bv)].$$(3) Clearly, the ranges of the Spearman’s rho, ${\rho }^{\ast },$ and Kendall’s tau, ${\tau }^{\ast },$ for the FGM family (3) are respectively $(-0.333,0.333)$ and $(-0.222,0.222).$ On the other hand, $$C(u,v)\le C^{\ast }(u,v), \text{for}\mathrm{ }\text{all}\mathrm{ }\text{vlaues} 0\le u,v\le 1.$$ Thus, $\rho \le {\rho }^{\ast }<0.333$ and $\tau \le {\tau }^{\ast }<0.222.$