Bivariate q-extended Weibull morgenstern family and correlation coefficient formulas for some of its sub-models

1.   Introduction

Let $(X, Y)$ be a pair of absolutely continuous random variables (r.v.'s), formally defined by the cumulative distribution function (DF)

where $F_X(x)$ and $F_Y(y)$ are the marginals DF's of $X$ and $Y,$ respectively. Moreover, the two kernels $A(x)\to0$ and $B(x)\to0,$ as $x\to 1,$ satisfy certain regularity conditions ensuring that $F_{X, Y}(x, y)$ is a DF with absolutely continuous marginals $F_X(x)$ and $F_Y(y).$ Usually, any bivariate distribution $F_{X, Y}(x, y)$ that has the above representation is referred to as FGM distribution. This model was originally introduced by Morgenstern (1956)^[11] for $A(x) = 1-x$ and $B(y) = 1-y$ and investigated by Gumbel (1960)^[7] for exponential with its multivariate case. The admissible range of association parameter $\lambda$ for the distribution, with $A(x) = 1-x$ and $B(y) = 1-y,$ is $-1\leq\lambda\leq1$ and the Pearson correlation coefficient between $X$ and $Y$ can never exceed 1/3. The classical FGM distribution is a flexible family useful in applications provided that the correlation between the variables is not too large. It can be utilized for arbitrary continuous marginals $F_X(x)$ and $F_Y(y).$

The maximum correlation coefficient is 1/4 when the marginal distributions are exponential (Gumbel, 1960^[7]). The maximum correlation coefficients equal 0.2921 and 0.314 for generalized exponential (Tahmasebi and Jafari, 2015^[17]) and Rayleigh distributions (Tahmasebi et al., 2018^[18]), restrictively.

In mathematical physics and probability, the q-distribution is more general than classical distribution. It was introduced by Diaz and Parguan (2009)^[4] and Diaz et al. (2010)^[3] in the continuous case and by Charalambides (2010)^[1] in the discrete case. The construction of a q-distribution is the construction of a q-analogue of ordinary distribution. Mathai (2005)^[10] introduced the q-analogue of the gamma distribution with respect to Lebesgue measure. Recently, several q-type super statistical distributions, such as the q-exponential, q-Weibull and q-logistic, were developed in the context of statistical mechanics, information theory and reliability modelling as discussed for instance by Sol and Kac (2005)^[15] and Srivastave and Choi (2012)^[16]. Provost et al. (2018)^[14] introduced the q-generalized extreme value under linear normalization (q-GEVL) and q-Gumbel distributions.

Gurvich et al. (1997)^[8] introduced a class of extended Weibull family (EWF) of distributions with survival function (SF),

where $\eta$ is a vector of parameters and $\Psi (x; \eta)$ is a non-negative, continuous, monotone increasing and differentiable function of $x$, which depends on the parameter vector $\eta$. Also, $\Psi (x; \eta)\rightarrow 0^{+}$ as $x\rightarrow 0^{+}$ and $\Psi (x; \eta)\rightarrow \infty$ as $x\rightarrow \infty.$ The probability density function (PDF) corresponding to (1.3) can be listed as

where $\Psi^{(1)}(x, \eta)$ is the derivative of $\Psi (x; \eta)$ with respect to $x$.

Let $X$ be a continuous random variable from the new family which is introduced by Nigm and Sadk (2021a). This family is called q-extended Weibull family and denoted by q-EWF. The SF is:

where $\eta$ could be a vector of parameters and $\Psi(x; \eta)$ is non-negative, monotonically increasing function of $x$, not depending on $\theta$ and differentiable with respect to $x$. The PDF corresponding to (1.5) is:

Note that if $q\rightarrow 0$ in (1.5) and (1.6) we have (1.3) and (1.4).

Some special cases of the q-EWF with the corresponding $\Psi(x; \eta)$ are listed in Table 1.

The inference for q-EWF were considered under Type-II progressive censoring with random removal by Nigm and Sadk (2021b)^[12]. Also, a class of models called Marshall-Olkin q-extended Weibull family was proposed by Nigm and Sadk (2021c)^[13]. There are a lot of papers on the same topic (Cousineau (2009)^[2], Gauchman (2004)^[5], Gradshteyn and Ryzhik (2007)^[6] and Krugman (1996)^[9]).

In this paper, we propose a bivariate extension of the q-EWF using the FGM approach. The main topic is finding the maximum correlation coefficient for some special cases of the FGM q-EWF. For some distributions, the correlation coefficient can be obtained mathematically when the $r_{th}$ moment of the marginal distribution exists. Otherwise, the correlation coefficient can be obtained numerically. The paper is organized as follows: In Section 2, some properties of the FGM q-EWF distribution are investigated. In Section 3, some special cases of this family are studied and the maximum correlation is obtained. Finally, some conclusion remarks are reported in Section 4.

2.   FGM q-EWF $(\theta, \eta, q)$ model

The joint CDF and PDF of $(X, Y)$ are defined by (1.1) and (1.2), respectively, where $X\sim \mbox{q-EWF}(\theta_1, \eta_1, q_1)$ and $Y\sim \mbox{q-EWF}(\theta_2, \eta_2, q_2).$ Thus, it is easy to show that the $(n, m)$th joint moment of the FGM q-EWF $(\lambda, \theta_1, \theta_2, q_1, q_2)$ family is given by

where $U_{1}\sim \mbox{q-EWF} (2\theta_1;\eta_1, q_{1}/{2})$ and $U_{2}\sim \mbox{q-EWF}(2\theta_2;\eta_2, q_{2}/{2}).$

Also, the covariance of $(X, Y)$ is

The correlation coefficient of $(X, Y)$ is

3.   Some special cases of the FGM q-EWF $(\theta, \eta, q_i)$

In this section, we obtain the maximal correlation coefficient for special cases of the FGM q-EW$(\theta, \eta, q)$ family distribution through the following sub-sections:

3.1.   q-Uniform distribution (q-U$(\theta, \eta, q)$)

The q-U$(\theta, \eta, q)$ model can be expressed by putting $\Psi(x, {\eta }_1) = -\mathrm{log}(1-x); 0 < x < 1, \; \theta = 1$ in (1.5) and (1.6). Figure 1 shows the PDF, SF and Hazard function (HF) of q-U$(\theta, \eta, q)$ for various values of the parameter $q$. According to Figure 1, it is noted that q-U $(\theta, \eta, q_i)$ cannot be used to model left and right skewed data.

The following theorem lists the new created model based on the coefficient of correlation.

Theorem 3.1. Let $(X, Y) \sim$ bivariate FGM q-U $(\theta_i, \eta_i, q_i),$ for $\Psi(x, {\eta}_1) = -\mathrm{log}(1-x), \; \; \Psi(y, {\eta }_2) = -\mathrm{log}(1-y), \; 0 < x, y < 1, \; {\theta }_i = 1,$ and $U_i\ \sim q-U(2\theta _{i}, \eta _{i}, q_{i}/{2}), \; \; i = 1, 2.\;$ Then

where

and

Proof. With $\Psi(x, \eta_1) = -\log(1-x), \; \theta_1 = 1$ in (1.5) and (1.6), we have

Then

Put $t = -\log{(1-x)}$, we get

where $\varphi_k(q_1, t)$ defined in (3.3). Similarly, we have

where $\varphi_j(q_2, t)$ defined in (3.3). Also, $U_i\ \sim q-U(2\theta _{i}, \eta _{i}, \frac{q_{i}}{2}), \; i = 1, 2$ then:

where $\xi_k(q_1, t)$ defined in (3.4) and

where $\xi_j(q_2, t)$ defined in (3.4).

From (3.6) and (3.7) for $n = m = 1, 2$, we get

From (3.8) and (3.9) for $n = m = 1$ we get

From (3.10), (3.12) and (3.13) in (2.3) we get

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Remark 3.1. If $q_i\rightarrow 0, i = 1, 2$ we see that $\varphi_1(q_i, t) = \frac{1}{2}$, $\varphi_2(q_i, t) = \frac{1}{3}$, $\xi_{1}(q_i, t) = \frac{1}{3}$ and $\rho (X, Y) = \frac{\lambda}{3}$.

The following table (see Table 2) gives the values of the correlation coefficient in case FGM q-Uniform distribution.

3.2.   q-exponential distribution (q-Exp$(\theta, \eta, q)$)

By putting $\Psi(x, {\eta }) = x\; , x > 0$ in (1.5), we will have $q-Exp(\theta, \eta, q)$. Figure 2 illustrates the PDF, the SF and HF for q-exponential distribution for $q = 0.3$ and different values of $\theta$ as follow:

We obtain the maximum correlation coefficient from special cases of FGM q-EWF $(\theta, \eta, q)$ distribution [$q-Exp(\theta, \eta, q)$] through the following theorem.

Theorem 3.2. Let $(X, \; Y) \sim$ bivariant FGM q-Exp $(\theta_i, \eta_i, q_i),$ for $\Psi(x, \eta _1) = x, \Psi(y, \eta _2) = y, 0 < x, y < \infty, \; \theta _i > 0$ $, \; U_i\sim \; \; q$-$Exp (2\theta _{i}, \eta _{i}, \frac{q_i}{2}), \; i = 1, 2$. Then

and

Proof. With $\Psi(x, \eta _1) = x$ in (1.5) and (1.6) we have,

Then

Put $t = q_1\theta_{1}x$

we get

Let $u = t/(t+1)$

then,

Similarly, we have

Also, $\; U_i \sim q$-$Exp(2\theta _{i}, \eta _{i}, q_i/{2}), \; i = 1, 2$

then:

and

From (3.18) and (3.19) for $n = m = 1, 2$ we get

From (3.20) and (3.21) for $n = m = 1$ we get

From (3.22)–(3.24) in (2.3) we get

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Remark 3.2. If $q_i\rightarrow 0, i = 1, 2,$ we see that $E(X) = 1/{\theta_1}, \; \; \; E(Y) = 1/{\theta_2}, Var(X) = 1/{\theta_1^{2}}, \; Var(Y) = 1/{\theta_2^{2}}, E(U_1) = 1/{2\theta_1}, \; \; \; \; E(U_2) = 1/{2\theta_2}\; \; and\; \; \rho (X, Y) = \lambda/ {4}$

The following tables (see Table 3) gives the values of the correlation coefficient in case FGM q-exponential distribution.

3.3.   q-Weibull distribution (q-W$(\theta, \eta, q))$)

By putting $\Psi(x, {\eta }) = x^{b}\; ,x > 0$ in (1.5), we have $q$-$Weibull(\theta, \eta, q)$. Figure 3 illustrates the probability density function, survival function, hazard function of q-Weibull distribution for $q = 0.3$ and some different values of $\theta$ and $b$ as follows:

From Figures 2 and 3, we see that the q-exponential and q-Weibull distributions can be used for modeling various data in different fields. The previous line of the PDF of the two distributions can be unimodal and bimodal. The hazard function for these distributions have various shapes, including increasing, decreasing, unimodal or bathtub.

The maximum correlation coefficient from special cases of FGM q-EWF $(\lambda, \theta, q)$ family distributions [$q-W(\theta, \eta, q)$] will be obtained through the following theorem.

Theorem 3.3. Let $(X, Y) \sim$ bivariant FGM q-W $(\theta_i, \eta_i, q_i),$ for, $\Psi(x, \eta _1) = x^{b_1}, Psi(y, \eta_2) = y^{b_2}, \; 0 < x, y < \infty, \; b_i > 0, \; \theta_i > 0, \; U_i\sim$ q-W $(2\theta_{i}, \eta_{i}, q_i/{2}), i = 1, 2.$ Then

and

Proof. With $\Psi(x, \eta _1) = x^{b_1} \; in\; (1.5)$ and $(1.6)$ we have

Then

Put $t = q_1\theta_{1}x^{b_1}$

let $u = t/{t+1}$

then

Similarly, we have

Also, $U_i \sim q-W(2\theta _{i}, \eta _{i}, q_i/{2}), \; i = 1, 2$

then:

and

From (3.29) and (3.30) for $n = m = 1, 2$. we get

From (3.31) and (3.32) for $n = m = 1$ we get

From (3.33)–(3.35) in (2.3) we get

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Remark 3.3. From Theorem 3.2:

(1) If $q_i\rightarrow 0, i = 1, 2$ we see that,

then,

(2) If $b_1 = b_2 = 1$ then q-Weibull distribution becomes q-exponential distribution (as in Theorem 3.2).

The following table (see Table 4) gives the values of the correlation coefficient in case FGM q-Weibull distribution.

3.4.   q-pareto distribution (q-pareto$(\theta, \eta, q)$)

By putting $\Psi(x, {\eta }) = \log(x/{b})$ in (1.5) $, \; \eta = b, \; x\geq b$ we will have $q-pareto(\theta, b, q)$. Figure 4 illustrates the PDF, the SF and HF for q-pareto distribution for $q = 1$ and different values of $\theta$ and $b$ as follow:

The maximum correlation coefficient from special case of FGM q-pareto $(\theta, \eta, q)$ distribution [$q-pareto(\theta, b, q)$] will be obtained through the following theorem.

Theorem 3.4. Let $(X, \; Y) \sim$ bivariant FGM q-pareto $(\theta_i, b_i, q_i),$ for $\Psi(x, \eta _1) = \log(x/{b_1}), \Psi(y, \eta _2) = \log(y/{b_2}), \; x\geq b_1, \; \; y\geq b_2, \; \theta _i > 0$ $, \; U_i\sim \;$ q-pareto $\; (2\theta _{i}, b _{i}, q_{i}/{2}), \; i = 1, 2$. Then

where,

$\varphi_k(t; q_1, \theta_1) = \theta_1\int_{0}^{\infty}t^k(1+q_1\theta_1t)^{-\frac{1}{q_1}-1}dt, \; \; \; \varphi_j(t; q_2, \theta_2) = \theta_2\int_{0}^{\infty}t^j(1+q_2\theta_2t)^{-\frac{1}{q_2}-1}dt,$

$\varphi_{k}^{*}(t; q_1, \theta_1) = \theta_1\int_{0}^{\infty}t^k(1+q_1\theta_1t)^{-\frac{2}{q_1}-1}dt, \; \; \; \varphi_{j}^{*}(t; q_2, \theta_2) = \theta_2\int_{0}^{\infty}t^j(1+q_2\theta_2t)^{-\frac{2}{q_2}-1}dt, \; \;$

and

Proof. With $\Psi(x, \eta _1) = \log(\frac{x}{b_1})$ in (1.5) and (1.6) we have,

Then

Put $t = \log{x/{b_1}}$ we get

Then,

where, $\varphi_k(t; q_1, \theta_1) = \theta_1\int_{0}^{\infty}t^k(1+q_1\theta_1t)^{-\frac{1}{q_1}-1}dt$.

Similarly, we have

where, $\varphi_j(t; q_2, \theta_2) = \theta_2\int_{0}^{\infty}t^j(1+q_2\theta_2t)^{-\frac{1}{q_2}-1}dt$.

Also, $U_i \sim q-pareto(2\theta _{i}, \eta _{i}, \frac{q_i}{2}),$ then,

where, $\varphi_{k}^{*}(t; q_1, \theta_1) = \theta_1\int_{0}^{\infty}t^k(1+q_1\theta_1t)^{-\frac{2}{q_1}-1}dt$

and

where, $\varphi_{j}^{*}(t; q_2, \theta_2) = \theta_2\int_{0}^{\infty}t^j(1+q_2\theta_2t)^{-\frac{2}{q_2}-1}dt$.

From (3.18) and (3.41) for $n = m = 1, 2$ we get

then,

From (3.19) and (3.43) for $n = m = 1$ we get

From (3.44)–(3.46) in (2.3) we get

□

Remark 3.4. If $q_i\rightarrow 0, i = 1, 2,$ we see that

for $\theta_i\rightarrow \infty$ then $|\rho(X, \; Y)| < \frac{1}{4}$.

Table 5 gives the values of the correlation coefficient in case FGM q-pareto distribution.

From Tables 2–5, we see that the maximum values of the correlation coefficient for the four models, FGM q-Uniform distribution, FGM q-exponential distribution, FGM q-Weibull distribution and FGM q-pareto distributionare, are different according the values of the parameters $q_{1}$ and $q_2$.

Table 2 shows that the maximum value of the correlation coefficient $\rho = 0.333$ when $q_1$ and $q_2$ with negative and positive values approach zero.

From Table 3, we have the maximum correlation coefficient value $0.645 \leq \rho \leq 0.953$ when $4\leq q_1, q_2 \leq 10$.

Table 4 shows that the maximum correlation coefficient $\rho = 0.3218$ when $q_1$ and $q_2$ take some positive values, which approaches zero. From this table, we also see that the values of $\rho$ increases when $b_1$ and $b_2$ increase. In Table 5, we have the maximum correlation coefficient value $0.53 \leq \rho \leq 0.9389$ for different values of $q_1$ and $q_2$. The values of $\theta_1$ is considered a pivotal role in increasing the values of correlation coefficient. Finally, the values of $q_1$ and $q_2$ play important roles to increase the maximum value of the correlation coefficient. Also, the values of $q_1$ and $q_2$ are not unique for the four models to increase the maximum correlation coefficient.

4.   Conclusions

In this paper, we propose the bivariate extension of the q-EWF using the FGM approach. The main topic is finding the maximum correlation coefficient for some special cases of the FGM q-EWF. For some distributions, the correlation coefficient was obtained mathematically when the $r^{th}$ moment of marginal distribution exists. Otherwise, the correlation coefficient is obtained numerically.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

Researchers would like to thank the Deanship of Scientific Research, Qassim University for funding publication of this project.

Conflict of interest

The authors declare no conflict of interest.