Bivariate q- generalized extreme value distribution: A comparative approach with applications to climate related data

Abstract

The premise of extreme value theory focuses on the stochastic behaviour and occurrence of extreme observations in an event that is random. Traditionally for univariate case, the behaviour of the maxima is described either by the types-I, types-II or types-III extreme value distributions, primarily known as the Gumbel, Fréchet or reversed Weibull models. These are all particular cases of the generalized extreme value ($\mathcal{G}\mathcal{E}\mathcal{V}$) model. However, in real-world scenario, these incidents take place as a consequence of concurrent dependent random events, where the relationship between the two variables is unidirectional or asymmetrical. [1] introduced a rigorous univariate extension of $\mathcal{G}\mathcal{E}\mathcal{V}$ distribution involving an additional parameter, the q− generalized extreme value ($q-\mathcal{G}\mathcal{E}\mathcal{V}$) distribution, as well as the q− Gumbel distribution. The prime interest of this paper lies in conceptualizing a novel approach to model bi-variate ($\mathcal{E}\mathcal{V}$) data, arising naturally from independent $q-\mathcal{G}\mathcal{E}\mathcal{V}$ random variables. This is achieved via the transformation of variables technique by establishing the resulting supports. Concisely, a technique is developed to model interdependent bivariate observations consisting of extreme values in terms of $q-\mathcal{G}\mathcal{E}\mathcal{V}$ probability density functions. Besides, we employed the suggested technique to a bivariate flood data set and demonstrate the competitiveness of the proposed bivariate $q-\mathcal{G}\mathcal{E}\mathcal{V}$. Additionally, conventional method to propose the newly defined bivariate ($q-\mathcal{G}\mathcal{E}\mathcal{V}$) distribution with bivariate q− Gumbel distribution (a special case for $\xi \to 0$) has also been established with related inferences and application to climate data.

Nomenclature

GEV

generalized extreme value

q−GEV

q− generalized extreme value distribution

bi-variate$\mathcal{E}\mathcal{V}$

bi-variate extreme value

CDF

Cumulative Distribution Function

PDF

Probability Density Function

MLE'

Maximum Likelihood Estimate

BI$q-\mathcal{G}\mathcal{E}\mathcal{V}$

Conventional bivariate q-generalized extreme value

BIq-Gumbel

Conventional bivariate q-Gumbel distribution

JCDF

Joint CDF of BI$q-\mathcal{G}\mathcal{E}\mathcal{V}$

AIC

Akaike Information Criterion

CAIC

Corrected Akaike Information Criterion

BIC

Bayesian Information Criterion

HQIC

Hannan-Quinn Information Criterion

1. Introduction

Long-lasting changes to earth's climatic patterns, such as adjustments to temperature, precipitation, and atmospheric conditions, are collectively referred to as climate change. Over the past century, human activities, particularly the combustion of petroleum and other fossil fuels and logging, have substantially boosted the levels of emissions of greenhouse gases in the atmosphere. More heat is trapped by this strengthened greenhouse effect, causing global warming and ensuing modifications to weather cycles. The repercussions of climate change are vast spanning ecological concerns, extreme weather events, accelerating sea levels, and disturbances to communities. Encouraging sustainable habits, regulatory measures, and multilateral attempts to reduce the negative consequences of climate change and adapt to changing environmental circumstances have made addressing it a vital global priority. In order to secure a sustainable future for the world, there is an urgency to address climate change effects encompassing collective policies to the interdependence of ecological, social, and economic systems.

The aspect of triggered frequent occurrences of extreme events such as earthquakes, floods, hurricanes, droughts etc in a variety of scientific domains involving time-dependent hazards is colossal, prompting an increase in attention to study non-stationary stochastic processes. It comprises of the statistical modelling of extremely unlikely occurrences at some threshold level (maxima or minima) alongside the assessment of tail-related risk measures thereafter. In univariate case, generally the asymptotic behaviour of the maxima is characterized by $\mathcal{G}\mathcal{E}\mathcal{V}$ model defined on unbounded and bounded support for type-I (Gumbel), type-II (Fréchet) and type-III (reversed Weibull) distributions, respectively. The cumulative distribution function (CDF) and the probability density function (PDF) of the $\mathcal{G}\mathcal{E}\mathcal{V}(m,s,\xi )=\mathcal{G}\mathcal{E}{\mathcal{V}}^{⁎}(\mu ,\sigma ,\xi )$ distributions is given, respectively, as

$$F(x;s,m,\xi )=\{\begin{array}{cc}{\mathrm{e}}^{-{[1+\xi (sx-m)]}^{-1/\xi }},\hfill & \xi \ne 0\hfill \\ {\mathrm{e}}^{{\mathrm{e}}^{-{(sx-m)}^{-1/\xi }}},\hfill & \xi \to 0\hfill \end{array}$$ (1)

and the PDF is given by

$$f(x;s,m,\xi )=\{\begin{array}{cc}s{[\xi (-m+sx)+1]}^{-\frac{1}{\xi }-1}{\mathrm{e}}^{-1+\xi {(-sx+m)}^{-1/q}},\hfill & \xi \ne 0\hfill \\ s{\mathrm{e}}^{[-{\mathrm{e}}^{-(-m+sx)}]}{\mathrm{e}}^{-(m-sx)},\hfill & \xi \doteq 0\hfill \end{array}$$ (2)

with $m=\mu /\sigma$ as a location parameter; $s=\frac{1}{\sigma }$ as scale parameter and ξ as the power parameter, having following support of the distributions defined on Eq. (3) as:

$$x\in \{\begin{array}{cc}(\frac{m}{s}-\frac{1}{{\xi }_s},\infty ),\hfill & \xi >0,\hfill \\ (-\infty ,\infty ),\hfill & \xi \to 0,\hfill \\ (-\infty ,\frac{m}{s}-\frac{1}{{\xi }_s}),\hfill & \xi <0.\hfill \end{array}$$ (3)

The power parameter ξ influences the tail behaviour of the $\mathcal{E}\mathcal{V}$ distribution, where there is virtually little chance that anything will transpire. There are certain fields of application where the $\mathcal{G}\mathcal{E}\mathcal{V}$ distribution is referred as the Fisher-Tippet [2] distribution, named after Ronald Fisher and L. H. C. Tippet who scripted the three distinctive forms. Joints with collar plates operating at normal temperature or at a higher temperature may attain their optimum durability by utilizing the density of the $\mathcal{G}\mathcal{E}\mathcal{V}$ distribution was examined by the authors in [3]. Yet $\mathcal{G}\mathcal{E}\mathcal{V}$ are often restricted to employing Gumbel distribution as a predominant model to quantify extreme events. In [4], the authors employed the Gumbel distribution in modelling the score statistics of global sequence alignment. The authors in [5] studied the health implications of climate-related shifts in extreme event exposure, a scientific domain previously unexplored in contemporary literature. In [6], the authors produced a detailed review focussing the applications of Gumbel distribution in real world scenario. For further reading, the readers are referred to the work in [7], [8], [9], [10], [11], [12]. The functional form for all three distributions was discovered by [13]. The sub-families defined by $\xi =0$, $\xi >0$ and $\xi <0$ corresponds to the Gumbel, Fréchet and Reversed Weibull families, respectively. The corresponding CDFs of these families are shown in Eq. (4), (5) and (6) as follows

• When $\xi =0$, Gumbel or type-I extreme value distribution

$$F(x;s,m,0)={\mathrm{e}}^{-{\mathrm{e}}^{-(sx-m)}}forx\in \Re$$ (4)

• For $\xi =\frac{1}{\sigma }>0$ and $y=1+\xi (sx-m)$, Fréchet or type-II extreme value distribution

$$F(y;s,m,\alpha )=\{\begin{array}{cc}{\mathrm{e}}^{{(-y)}^{\alpha }},\hfill & y>0,\hfill \\ 0\hfill & y\le 0\hfill \end{array}$$ (5)

• If $\xi =\frac{1}{\sigma }<0$ and $y=[1+\xi (sx-m)]$, Reversed Weibull or type-III extreme value distribution

$$F(y;s,m,\alpha )=\{\begin{array}{cc}{\mathrm{e}}^{-{(-y)}^{\alpha }},\hfill & y<0,\hfill \\ 1\hfill & y\ge 0\hfill \end{array}$$ (6)

Remark 1

The theory here relates to maxima and the distribution being discussed is an extreme value distribution for maxima. The $\mathcal{G}\mathcal{E}\mathcal{V}$ distribution for minima can be obtained, for example by substituting (-x) for x in the distribution functions and subtracting from one. The $\mathcal{G}\mathcal{E}\mathcal{V}$ distribution is frequently employed to represent the risk of extreme, rare events in finance (to model jumps in the stock market), insurance (to determine the distribution of claims), hydrology (for predicting flood levels), seismology and other fields of scientific investigation including economics, material sciences, telecommunications, engineering and time series modelling. Most recent application in structural engineering includes the adoption of $\mathcal{G}\mathcal{E}\mathcal{V}$ model to determine the maximum capacity values of X-joints reinforced with outer rings at low and high temperatures by [14]; to explore the stress-strength relationship of tube joints in out-of-plane and in-plane bending loads in [15] and [16] among others. Here are some relationships involving the $\mathcal{G}\mathcal{E}\mathcal{V}$ distribution:

• •If $X\sim \mathcal{G}\mathcal{E}{\mathcal{V}}^{⁎}(\mu ,\sigma ,0)$ then $mX+b\sim \mathcal{G}\mathcal{E}{\mathcal{V}}^{⁎}(m\mu +b,m\mu ,0)$.
• •If $X\sim Gumbel(\mu ,\sigma )$ then $X\sim \mathcal{G}\mathcal{E}{\mathcal{V}}^{⁎}(\mu ,\sigma ,0)$.
• •If $X\sim Weibull(\sigma ,\mu )$ then $\mu (1-\sigma \log \frac{X}{\sigma })$$\sim \mathcal{G}\mathcal{E}{\mathcal{V}}^{⁎}(\mu ,\sigma ,0)$.
• •If $X\sim \mathcal{G}\mathcal{E}{\mathcal{V}}^{⁎}(\mu ,\sigma ,0)$ then $\sigma \exp (-\frac{X-\mu }{\mu \sigma })$$\sim Weibull(\sigma ,\mu )$.
• •If $X\sim Exponential($1) then $\mu -\sigma \log (X)\sim \mathcal{G}\mathcal{E}{\mathcal{V}}^{⁎}(\mu ,\sigma ,0)$
• •If $X\sim \mathcal{G}\mathcal{E}{\mathcal{V}}^{⁎}(\alpha ,\beta ,0)$, and $Y\sim \mathcal{G}\mathcal{E}{\mathcal{V}}^{⁎}(\alpha ,\beta ,0)$ then $X-Y\sim logistic(0,\beta )$.
• •If $X\sim \mathcal{G}\mathcal{E}{\mathcal{V}}^{⁎}(\alpha ,\beta ,0)$, and $Y\sim \mathcal{G}\mathcal{E}{\mathcal{V}}^{⁎}(\alpha ,\beta ,0)$ then $X+Y\sim logistic(2\alpha ,\beta )$.

For the basic distributional properties of the extreme value distributions and consideration on their numerous applications, the reader is referred to [17], [18], [19] and [20].

In practice, however, the $\mathcal{G}\mathcal{E}\mathcal{V}$ distribution is desired for modelling certain types of data sets with more flexible forms that ought computationally. These extended models are referred to as q-analogues of the distributions of origin, with an additional parameter denoted by q incorporated in their density functions. In the same vein, Provost et al. The $q-\mathcal{G}\mathcal{E}\mathcal{V}$ and q− Gumbel distributions, initially, was synthesized by the authors in [1]. The CDF and PDF of the $q-\mathcal{G}\mathcal{E}\mathcal{V}$ and q-Gumbel (letting $\xi \to 0$) distributions which generalize Eq. (1) and Eq. (2) are respectively given by

$$F(x;s,m,q,\xi )=\{\begin{array}{cc}{[1+q{\{\xi (-m+sx)+1\}}^{-1/\xi }]}^{-1/q},\hfill & \xi \ne 0,q\ne 0\hfill \\ {(1+q{\mathrm{e}}^{-(-m+sx)})}^{-1/q},\hfill & \xi \to 0,q\ne 0\hfill \end{array}$$ (7)

$$f(x;s,m,q,\xi )=\{\begin{array}{cc}s{[1+\xi (-m+sx)]}^{\frac{-1}{\xi }-1}{\{1+q{[\xi (-m+sx)+1]}^{-1/\xi }\}}^{\frac{-1}{q}-1},\hfill & \xi \ne 0,q\ne 0\hfill \\ s{\mathrm{e}}^{-sx+m}{(1+q{\mathrm{e}}^{-sx+m})}^{-\frac{1}{q}+1},\hfill & \xi \doteq 0,q\ne 0,\hfill \end{array}$$ (8)

where $x\in S_X^{(i)}$ for $i=1,2,...,6$ and $S_X^{(i)}$ is the support of the distribution as follows:

$$S_X^{(i)}=\{\begin{array}{cc}(\frac{m}{s}-\frac{1}{\xi s},\infty ),\hfill & \xi >0,q>0,\hfill \\ (-\infty ,\frac{m}{s}-\frac{1}{\xi s}),\hfill & \xi <0,q>0,\hfill \\ (\frac{{(-q)}^{\xi }-1}{\xi s}+\frac{m}{s},\infty ),\hfill & \xi >0,q<0,\hfill \\ (\frac{{(-q)}^{\xi }-1}{\xi s}+\frac{m}{s},\frac{m}{s}-\frac{1}{\xi s}),\hfill & \xi <0,q<0,\hfill \\ (-\infty ,\infty ),\hfill & \xi \to 0,q>0,\hfill \\ (\frac{m-\ln (-q)}{s},\infty ),\hfill & \xi \to 0,q<0.\hfill \end{array}$$ (9)

Much of the extreme value literature is dedicated to univariate case yet these events may be the outcome of two simultaneous events, casually termed as bi-variate random variables. For example, an abrupt vertical shift in the ocean floor usually triggers a tsunami; a sudden rise in atmospheric temperature may cause wildfires; an earthquake can cause a nuclear accident; a drought may affect the ecology in a region. In such instances, bi-variate distribution is aptly utilized to give probabilities for simultaneous outcomes of the two random variables which may be deemed as dependent. Relatively, only a handful amount of work in references [21], [22], [23], [24], [25], [26], [27], [28], has been conducted on bivariate-($\mathcal{E}\mathcal{V}$) theory or $q-(\mathcal{G}\mathcal{E}\mathcal{V}$) which is quite surprising. With the hope to bridge this literary void, we put our prime focus to introduce a bi-variate $q-\mathcal{G}\mathcal{E}\mathcal{V}$ distribution via a novel as well as classical approach. The article is articulated as: The theoretical foundations to constitute a novel bivariate q- extended distribution with bi-variate $\mathrm{q}-\mathcal{GEV}$ (a special case) distributions are presented in Section 2; further, the resultant model is applied on a real life data set related to flood frequency level (section 2.2); Section 3 comprises of conventional bi-variate extension of $q-\mathcal{G}\mathcal{E}\mathcal{V}$. The related inference regarding the parameters is conducted with the help of simulation analysis and the model is applied to climate data; Lastly, some conclusive remarks are summarized concisely in section 4 with final remarks.

2. Bivariate $q-\mathcal{G}\mathcal{E}\mathcal{V}$ distribution: a novel approach

We are hereby introducing the bivariate $q-\mathcal{G}\mathcal{E}\mathcal{V}$ and q− Gumbel distributions based on Eq. (7) and Eq. (8), respectively. First, we assume that the variables are independently distributed and then we consider the correlated case. The PDF of the $q-\mathcal{G}\mathcal{E}\mathcal{V}$ and q− Gumbel (obtained by letting $\xi \to 0$ in the $q-\mathcal{G}\mathcal{E}\mathcal{V}$ model) distributions are given by:

$$f_{i(x_j)}=f_i(x_j;s_j,m_j,{\xi }_j,q_j)=\{\begin{array}{c}{s}_j{(1+{\xi }_j(s_j{x}_j-m_j))}^{\frac{-1}{{\xi }_j}-1}{(1+q_j{({\xi }_j(s_j{x}_j-m_j)+1)}^{\frac{-1}{{\xi }_j}})}^{\frac{-1}{q_j}-1},\hfill \\ {\xi }_j\ne o,q_j\ne 0\hfill \\ s_j{\exp }^{m_j-s_j{x}_j}{(1+q_j{\exp }^{m_j-s_j{x}_j})}^{\frac{-1}{q_j}-1},\hfill & {\xi }_j\to 0,q_j\ne 0\hfill \end{array}$$

where $i=1,...,6$, depending on the support of the distribution, and $j=1,2$ (corresponding to the two components of the bivariate case being discussed) and $x_j\in S_{X_j}^{(i)}$ where $S_{X_j}^{(i)}$ is the support of the distributions as specified below:

$$S_{X_j}^{(i)}=\{\begin{array}{cc}(\frac{m_j}{s_j}-\frac{1}{{\xi }_j{s}_j},\infty ),\hfill & {\xi }_j>0,q_j>0,\text{ for }i=1,\hfill \\ (-\infty ,\frac{m_j}{s_j}-\frac{1}{{\xi }_j{s}_j}),\hfill & {\xi }_j<0,q_j>0,\text{ for }i=2,\hfill \\ (\frac{{(-q_j)}^{{\xi }_j}-1}{{\xi }_j{s}_j}+\frac{m_j}{s_j},\infty ),\hfill & {\xi }_j>0,q_j<0,\text{ for }i=3,\hfill \\ (\frac{{(-q_j)}^{{\xi }_j}-1}{{\xi }_j{s}_j}+\frac{m_j}{s_j},\frac{m_j}{s_j}-\frac{1}{{\xi }_j{s}_j}),\hfill & {\xi }_j<0,q_j<0,\text{ for }i=4,\hfill \\ (-\infty ,\infty ),\hfill & {\xi }_j\to 0,q_j>0,\text{ for }i=5,\hfill \\ (\frac{m_j-\ln (-q_j)}{s_j},\infty ),\hfill & {\xi }_j\to 0,q_j<0,\text{ for }i=6.\hfill \end{array}$$ (10)

2.1. Functional decomposition of $q-\mathcal{G}\mathcal{E}\mathcal{V}$ and q-Gumbel distributions

Constructing a theoretical bivariate distribution has always been a challenging problem, as it may involve some dependency between variables, which has to be accounted for. We now introduce a novel methodology in order to construct a bi-variate $q-\mathcal{G}\mathcal{E}\mathcal{V}$ distribution. Consider two independent variables $Z_1$ and $Z_2$ with $\sigma =1$ and $\mu =0$ and their PDFs denoted by $g_1(z_1)$ on $S_{Z_1}^{(i_1)}$ and $g_2(z_2)$ on $S_{Z_2}^{(i_2)}$ for $i_1,i_2=1,2,...,6$ and apply the transformation:

$$\left(\begin{array}{c}\hfill x_1\hfill \\ \hfill x_2\hfill \end{array}\right)={\left(\begin{array}{cc}\hfill {\sigma }_{11}\hfill & \hfill {\sigma }_{12}\hfill \\ \hfill {\sigma }_{12}\hfill & \hfill {\sigma }_{22}\hfill \end{array}\right)}^{\frac{1}{2}}\cdot \left(\begin{array}{c}\hfill z_1\hfill \\ \hfill z_2\hfill \end{array}\right)+\left(\begin{array}{c}\hfill {\theta }_1\hfill \\ \hfill {\theta }_2\hfill \end{array}\right)$$ (11)

where ${\sigma }_{j_1{j}_2}=E[(X_{j_{{}_1}}-E(X_{j_{{}_1}}))(X_{j_2}-E(X_{j_2}))],for{j}_1,j_2=1,2$, and ${\left(\begin{array}{cc}\hfill {\sigma }_{11}\hfill & \hfill {\sigma }_{12}\hfill \\ \hfill {\sigma }_{12}\hfill & \hfill {\sigma }_{22}\hfill \end{array}\right)}^{\frac{1}{2}}$ =$\left(\begin{array}{cc}\hfill {\alpha }_{11}\hfill & \hfill {\alpha }_{12}\hfill \\ \hfill {\alpha }_{12}\hfill & \hfill {\alpha }_{22}\hfill \end{array}\right)$

Letting

$$\mathrm{\Sigma }\equiv \left(\begin{array}{cc}\hfill {\sigma }_{11}\hfill & \hfill {\sigma }_{12}\hfill \\ \hfill {\sigma }_{12}\hfill & \hfill {\sigma }_{22}\hfill \end{array}\right)$$

denotes the covariance matrix of $X={(X_1,X_2)}^{\prime }$, the inverse transformation is

$$\left(\begin{array}{c}\hfill z_1\hfill \\ \hfill z_2\hfill \end{array}\right)=\mathrm{\Sigma }{}^{-1/2}\cdot \left(\begin{array}{c}\hfill x_1-{\theta }_1\hfill \\ \hfill x_2-{\theta }_2\hfill \end{array}\right)$$ (12)

One has $z_1={\beta }_{11}(x_1-{\theta }_1)+{\beta }_{12}(x_2-{\theta }_2)$ and $z_2={\beta }_{12}(x_1-{\theta }_1)+{\beta }_{22}(x_2-{\theta }_2)$.

Due to Eq. (12), the resulting correlated bivariate density function is

$$h(x_1,x_2)=g_1({\beta }_{11}(x_1-{\theta }_1)+{\beta }_{12}(x_2-{\theta }_2))g_2({\beta }_{12}(x_1-{\theta }_1)+{\beta }_{22}(x_2-{\theta }_2))\left|{\mathrm{\Sigma }}^{-1/2}\right|.$$ (13)

The density defined in Eq. (13) on the image of $S_{Z_1}^{i_1}\cdot S_{Z_2}^{i_2}$ is denoted by $S_X$ throughout the manuscript.

Case 1: This case the original support is $S_Z\equiv S_{Z_1}^{i_1}\cdot S_{Z_2}^{i_2}=(a_1,\infty )\cdot (a_2,\infty )$ with $i_1,i_2=1,3,6$ in Eq. (9).

When ${\sigma }_{j_1{j}_2}\ne 0$ for $j_1,j_2=1,2$ and the determinant of the matrix, ${\sigma }_{11}{\sigma }_{12}-{\sigma }_{{12}^2}\ne 0$; then the line $z_1=a_1$ and $z_2>b_1$ can be represented as $l_1$ and we have the following equations

$$\begin{gathered} x_1={\alpha }_{11}{a}_1+{\alpha }_{12}{a}_2+{\theta }_1 \\ {x}_2={\alpha }_{12}{a}_1+{\alpha }_{22}{a}_2+{\theta }_2 \end{gathered}$$

These equations are equivalent to

$$\frac{{\alpha }_{22}}{{\alpha }_{12}}{x}_1=\frac{{\alpha }_{22}}{{\alpha }_{12}}{\alpha }_{11}{a}_1+{\alpha }_{22}{z}_2+\frac{{\alpha }_{22}}{{\alpha }_{12}}{\theta }_1$$ (14)

$$x_2={\alpha }_{12}{a}_1+{\alpha }_{22}{z}_2+{\theta }_2$$ (15)

On subtracting Eq. (14) from Eq. (15), we obtain the transformed line

$$\widetilde{l_1}:x_2=\frac{{\alpha }_{22}}{{\alpha }_{12}}{x}_1+({\alpha }_{12}-\frac{{\alpha }_{22}}{{\alpha }_{12}}{\alpha }_{11})a_1+{\theta }_2-\frac{{\alpha }_{22}}{{\alpha }_{12}}{\theta }_2+{\alpha }_{12}{\theta }_1$$

Using the same procedure to find the transformed line when $z_2=b_1$ and $z_1>a_1$ which is represented by $l_2$, we have

$$\widetilde{l_2}:x_2=\frac{{\alpha }_{12}}{{\alpha }_{11}}{x}_1+({\alpha }_{22}-\frac{{\alpha }_{12}^2}{{\alpha }_{11}})b_1+{\theta }_2-\frac{{\alpha }_{12}}{{\alpha }_{11}}{\theta }_1$$ (16)

To determine the transformed intersection point $(\widetilde{a_1},\widetilde{b_1})$, we let Eq. (15)= Eq. (16), and then

$\widetilde{a_1}:x_1={\alpha }_{11}{a}_1+{\alpha }_{12}{b}_1+{\theta }_1$

$\widetilde{b_1}:x_2={\alpha }_{12}{a}_1+{\alpha }_{22}{b}_1+{\theta }_2$

Thus, under the transformation Eq. (11), the transformed region $S_X$ is bounded by the two lines

$l_1:z_1=a_1$ and $z_2>b_1\to \widetilde{l_1}:x_2=\frac{{\alpha }_{22}}{{\alpha }_{12}}{x}_1-({\alpha }_{11}{\alpha }_{22}-{\alpha }_{12}^2)\frac{a_1}{{\alpha }_{12}}+{\theta }_2-\frac{{\alpha }_{22}}{{\alpha }_{12}}{\theta }_1$ and $x_2>\widetilde{b_2}$ the intersection point being

$(\widetilde{a_1},\widetilde{b_1})=({\alpha }_{11}{a}_1+{\alpha }_{12}{b}_1+{\theta }_1,{\alpha }_{12}{a}_1+{\alpha }_{22}{b}_1+{\theta }_2)$.

We observe that the slope of $\widetilde{l_1}$ is $\frac{{\alpha }_{22}}{{\alpha }_{12}}$ and for $\widetilde{l_2}$ is $\frac{{\alpha }_{12}}{{\alpha }_{11}}$; therefore both slopes have the same sign which depends on the sign of ${\alpha }_{12}$. Fig. 1 presents the original support and the mapped supports depending on whether ${\alpha }_{12}$ is positive or negative.

Figure 1.

Original and transformed supports for Case 1 with positive and negative α₁₂.

Case 2: In this case, the original support is $S_Z\equiv S_{Z_1}^{(2)}\cdot S_{Z_2}^{(2)}=(-\infty ,a_2)\cdot (-\infty ,b_2)$, as defined in Eq. (10).

Applying the transformation Eq. (11), the region $S_X$ is bounded by the lines

$l_3:z_1=a_1$ and $z_2<b_1\to \widetilde{l_3}:x_2=\frac{{\alpha }_{22}}{{\alpha }_{12}}{x}_1-({\alpha }_{11}{\alpha }_{22}-{\alpha }_{12}^2)\frac{a_1}{{\alpha }_{12}}+{\theta }_2-\frac{{\alpha }_{22}}{{\alpha }_{12}}{\theta }_1$ and $x_2<\widetilde{b_1}$

$l_4:z_2=b_2$ and $z_1<a_1\to \widetilde{l_4}:x_2=\frac{{\alpha }_{12}}{{\alpha }_{11}}{x}_1+({\alpha }_{11}{\alpha }_{22}-{\alpha }_{12}^2)\frac{b_2}{{\alpha }_{11}}+{\theta }_2-\frac{{\alpha }_{12}}{{\alpha }_{11}}{\theta }_1$ and $x_2<\widetilde{a_2}$

and the intersection point is $(a_2,b_2)\to (\widetilde{a_2},\widetilde{b_2})$ =$({\alpha }_{11}{a}_2+{\alpha }_{12}{b}_2+{\theta }_2,{\alpha }_{12}{a}_2+{\alpha }_{22}{b}_2+{\theta }_2)$

If ${\alpha }_{ij}\ne 0$ for $i,j=1,2$ and the determinant of the matrix, ${\alpha }_{11}\cdot {\alpha }_{22}-{\alpha }_{12}^2\ne 0$. The resulting supports are shown in Fig. 2

Figure 2.

Original and transformed supports for Case 2 with positive and negative α₁₂.

If for ${\alpha }_{ij}\ne 0i,j=1,2$ and the determinant of the matrix, ${\alpha }_{11}\cdot {\alpha }_{22}-{\alpha }_{12}^2\ne 0$. The resulting supports are shown in Fig. 2.

Case 3: This case, the original support is $S_Z=S_{Z_1}^{(4)}\cdot S_{Z_2}^{(4)}=(a_1,a_2)\cdot (b_1,b_2)$, as defined in Eq. (10).

Applying the transformation Eq. (11), the region $S_X$ is bounded by the four lines and the four corners.

$l_1:z_1=a_1$ and $b_1<z_2<b_2\to \widetilde{l_1}:x_2$=$\frac{{\alpha }_{22}}{{\alpha }_{12}}{x}_1-({\alpha }_{11}\cdot {\alpha }_{22}-{\alpha }_{12}^2)\frac{a_1}{{\alpha }_{12}}+{\theta }_2-\frac{{\alpha }_{22}}{{\alpha }_{12}}{\theta }_1$ and $\widetilde{b_1}<x_2<\widetilde{a_2}$

$l_2:z_2=b_1$ and $a_1<z_1<b_2\to \widetilde{l_2}:x_2$=$\frac{{\alpha }_{12}}{{\alpha }_{11}}{x}_1+({\alpha }_{11}\cdot {\alpha }_{22}-{\alpha }_{12}^2)\frac{b_1}{{\alpha }_{11}}+{\theta }_2-\frac{{\alpha }_{12}}{{\alpha }_{11}}{\theta }_1$ and $\widetilde{a_1}<x_2<\widetilde{a_2}$

$l_3:z_1=a_2$ and $b_1<z_2<b_2\to \widetilde{l_3}:x_2$=$\frac{{\alpha }_{22}}{{\alpha }_{12}}{x}_1+({\alpha }_{11}\cdot {\alpha }_{22}-{\alpha }_{12}^2)\frac{a_2}{{\alpha }_{12}}+{\theta }_2-\frac{{\alpha }_{22}}{{\alpha }_{12}}{\theta }_1$ and $\widetilde{b_1}<x_2<\widetilde{b_2}$

$l_4:z_2=b_2$ and $a_1<z_1<a_2\to \widetilde{l_4}:x_2$=$\frac{{\alpha }_{12}}{{\alpha }_{11}}{x}_1+({\alpha }_{11}\cdot {\alpha }_{22}-{\alpha }_{12}^2)\frac{b_2}{{\alpha }_{11}}+{\theta }_2-\frac{{\alpha }_{12}}{{\alpha }_{11}}{\theta }_1$ and $\widetilde{a_1}<x_2<\widetilde{a_2}$

$(a_i,b_j)\to (\widetilde{a_i},\widetilde{b_j})$ =$({\alpha }_{11}{a}_i+{\alpha }_{12}{b}_j+{\theta }_1,{\alpha }_{12}{a}_i+{\alpha }_{22}{b}_j+{\theta }_2)$ where $i,j=1,2$.

The resulting four-sided supports are shown in Fig. 3.

Figure 3.

Original and transformed supports for Case 3 with positive and negative α₁₂.

Case 4: This case, the original support $Z_X\equiv S_{Z_1}^{(5)}\cdot S_{Z_1}^{(5)}=(-\infty ,\infty )\cdot (-\infty ,\infty )$, that is $i_1=i_2=5$ and the transformed support remains the entire real plane.

The cases discussed above are for both variables coming from the same support type. Now we consider different supports for each variable.

Case 5: In this case, the original support $Z_X=S_{Z_1}^{(5)}\cdot S_{Z_2}^{(1)}=(-\infty ,\infty )\cdot (b_1,\infty )$ that is $i_1=5$ $i_2=1,3,6$.

Applying the transformation Eq. (11), the region $S_X$ has only a bottom boundary that is represented by the line $\widetilde{l_1}$.

$l_1:z_2=b_1\to \widetilde{l_1}:x_2=\frac{{\alpha }_{12}}{{\alpha }_{11}}+({\alpha }_{11}\cdot {\alpha }_{22}-{\alpha }_{12}^2)\frac{b_1}{{\alpha }_{11}}+{\theta }_2-\frac{{\alpha }_{12}}{{\alpha }_{11}}{\theta }_1$.

The original and resulting supports are shown in Fig. 4.

Figure 4.

Original and transformed supports for Case 5 with positive and negative α₁₂.

Case 6: In this case, the original support is $Z_X\equiv S_{Z_1}^{(5)}\cdot S_{Z_2}^{(4)}=(-\infty ,\infty )\cdot (b_1,b_2)$, that is $i_1=5$ $i_2=4$.

Applying the transformation Eq. (12), the region $S_X$ is bounded by the two horizontal lines.

$l_1:z_2=b_1\to \widetilde{l_1}:x_2=\frac{{\alpha }_{12}}{{\alpha }_{11}}{x}_1+({\alpha }_{11}\cdot {\alpha }_{22}-{\alpha }_{12}^2)\frac{b_1}{{\alpha }_{11}}+{\theta }_2-\frac{{\alpha }_{12}}{{\alpha }_{11}}{\theta }_1$.

$l_2:z_2=b_2\to \widetilde{l_2}:x_2=\frac{{\alpha }_{12}}{{\alpha }_{11}}{x}_1+({\alpha }_{11}\cdot {\alpha }_{22}-{\alpha }_{12}^2)\frac{b_2}{{\alpha }_{11}}+{\theta }_2-\frac{{\alpha }_{12}}{{\alpha }_{11}}{\theta }_1$.

The original and resulting two-sided supports are shown in Fig. 5.

Figure 5.

Original and transformed supports for Case 6 with positive and negative α₁₂.

2.2. Modelling the bivariate flood data

Due to variations in temperature, radiation and wind speeds, climate change has a potent effect on how water supplies are transported and disseminated. Extreme hydrological occurrences, such as droughts and floods, are thus directly impacted by climate change. There are additional elements which influence runoff in a river basin, such as land use changes, water conservation initiatives and the expansion of water resources among others, that also alter regional water circulation. The authors in [29] explored an intriguing relationship of flood peak and flood volume in an interesting liaison of climatic and catchment processes. The data under consideration, studied by [30], comprises on $n=77$ of the maxima of the two variables observed during the period of 1990 to 1995 in the Madawaska basin, Quebec, Canada. Assume that $X={(X_1,X_2)}^{\prime }$ is a continuous bivariate random vector whose distribution has mean ${({\theta }_1,{\theta }_2)}^{\prime }$ and covariance matrix ∑. The standardized transformation is applied in order to remove the correlation between the two arbitrary variables as follows

$$\left(\begin{array}{c}\hfill z_1\hfill \\ \hfill z_2\hfill \end{array}\right)={\mathrm{\Sigma }}^{\frac{-1}{2}}\left(\begin{array}{c}\hfill x_1-{\theta }_1\hfill \\ \hfill x_2-{\theta }_2\hfill \end{array}\right),$$

where ${\mathrm{\Sigma }}^{\frac{-1}{2}}\equiv \left(\begin{array}{cc}\hfill {\beta }_{11}\hfill & \hfill {\beta }_{12}\hfill \\ \hfill {\beta }_{12}\hfill & \hfill {\beta }_{22}\hfill \end{array}\right)$ is the inverse of the symmetric square root of $\sum ^{ˆ}$ the estimated covariance matrix of X.

The product of the density estimates of $Z_1$ and $Z_2$, $g_1(z_1)g_2(z_2)$ serves an initial approximation.

The inverse transformation given by the following equation:

$$\left(\begin{array}{c}\hfill x_1\hfill \\ \hfill x_2\hfill \end{array}\right)={\mathrm{\Sigma }}^{\frac{1}{2}}\left(\begin{array}{c}\hfill z_1\hfill \\ \hfill z_2\hfill \end{array}\right)+\left(\begin{array}{c}\hfill {\theta }_1\hfill \\ \hfill {\theta }_2\hfill \end{array}\right)$$

is then applied and the resulting pdf estimate of the original variables is

$h(x_1,x_2)=g_1({\beta }_{11}(x_1-{\theta }_1))+g_2({\beta }_{12}(x_1-{\theta }_1))(({\beta }_{12}(x_1-{\theta }_1))+({\beta }_{22}(x_2-{\theta }_2))|{\sum }^{\frac{-1}{2}}|$ on $S_{X_1}^{(5)}\cdot S_{X_5}^{(5)}$.

The Fréchet, $\mathcal{G}\mathcal{E}\mathcal{V}$ and $q-\mathcal{G}\mathcal{E}\mathcal{V}$ (including the limiting case of q− Gumbel) models were fitted to the standardized variables in Fig. 6. The preferred model was identified as the q-Gumbel distribution for each variable whose support is $S^5$. The $ML{E}^{\prime }$ s of the parameters in Table 1 were acquired by maximizing the log-likelihood function using Mathematica.

Figure 6.

The histogram of marginal distribution of Z₁ (6-a) & Z₂ (6-b) with three fitted PDFs.

Table 1.

Estimation of the parameters for the bivariate Flood data.

Distribution	ML Estimates
$Fr\acute{e}chet(\alpha ,\beta ,\mu )$	${\hat{\alpha }}_1=1912.4$	${\hat{\beta }}_1=1479.56$	${\hat{\mu }}_1=-1478.36$
	${\hat{\alpha }}_2=184.905$	${\hat{\beta }}_2=143.352$	${\hat{\mu }}_2=-140.593$
$\mathcal{G}\mathcal{E}\mathcal{V}(s,m,\xi )$	${\hat{s}}_1=1.004$	${\hat{m}}_1=1.3$	${\hat{\xi }}_1=-0.283$
	${\hat{s}}_2=1.04$	${\hat{m}}_2=2.946$	${\hat{\xi }}_2=-0.225$
q − Gumbel (s,m,q)	${\hat{s}}_1=1.751$	${\hat{m}}_1=2.836$	${\hat{q}}_1=0.924$
	${\hat{s}}_2=1.436$	${\hat{m}}_2=4.311$	${\hat{q}}_2=0.487$

Table 2 comprises of standard model validation criterions including Akaike Information criterion (AIC), Corrected Akaike Information criterion (CAIC), Bayesian Information criterion (BIC) and Hannan-Quinn Information criterion (HQIC), respectively. These metrics indicate how effectively a model captures the underlying relationships and patterns in the data in comparison to other models. The visualizations in Figure 6, Figure 7 in conjunction with the empirical findings suggests that the proposed q-Gumbel model performs notably well as compared to the models. It is pertinent to note that while the proposed methodology might have its significance, one must keep in mind the fragile nature of the relationship between the two variable being linear, violation of which may lead to inaccurate modelling of the bivariate data. Therefore, the suggested approach may not be considered appropriate for capturing non-linear or intricate phenomena.

Table 2.

Fit indices for the bivariate flood data.

Distribution	AIC	AICC	BIC	HQIC
$Fr\acute{e}chet(\alpha ,\beta ,\mu )$	2329.84	2330.17	2336.87	2332.657
$\mathcal{G}\mathcal{E}\mathcal{V}(s,m,\xi )$	2337.33	2337.66	2344.36	2340.146
q-Gumbel (s,m,q)	2283.89	2284.22	2290.92	2286.705

Figure 7.

Kernel density plot of the original bivariate data (7-a) with joint density estimate obtained from the fitted $q-\mathcal{G}\mathcal{E}\mathcal{V}$ model (7-b) superimposed on a histogram of the original data.

3. Bivariate $q-\mathcal{G}\mathcal{E}\mathcal{V}$ : conventional approach

Recently, in references [31], [32], [33], [34], a more conventional approach has been adopted to introduce bivariate extensions of univariate models. The models defined through this framework are somewhat cumbersome. We label the bivariate $q-\mathcal{G}\mathcal{E}\mathcal{V}$ as Bi$q-\mathcal{G}\mathcal{E}\mathcal{V}$ distribution, in this section to differentiate from the previous section. For $\xi \to 0$, let us recall the $q-\mathcal{G}\mathcal{E}\mathcal{V}$ and q-Gumbel distributions in Eq. (1) and Eq. (2), respectively, as

$$F(x;s,m,q,\xi ))=\{\begin{array}{cc}{[1+q{\{\xi (-m+sx)+1\}}^{-(1/\xi )}]}^{-(1/q)},\hfill & \xi \ne 0,q\ne 0\hfill \\ {(1+q{\mathrm{e}}^{-m+sx})}^{-(1/q)}\hfill & \xi \doteq 0,q\ne 0\hfill \end{array}$$

and

$$f(x;s,m,q,\xi ))=\{\begin{array}{cc}s{\{1+\xi (-m+sx)\}}^{-\frac{1}{\xi }-1}{[1+q{\{1+\xi (-m+sx)\}}^{-\frac{1}{\xi }}]}^{-\frac{1}{q}-1},\hfill & \xi \ne 0,q\ne 0\hfill \\ s{\mathrm{e}}^{m-sx}{(1+q{\mathrm{e}}^{m-sx})}^{-\{1+(1/q)\}},\hfill & \xi \doteq 0,q\ne 0\hfill \end{array}$$

where $c,k,\alpha ,\beta >0$ and $x>0$.

3.1. Bi$q-\mathcal{G}\mathcal{E}\mathcal{V}$ and Biq-Gumbel models: core functions

Let $(X,Y)$ be a two dimensional random vector, $s,m>0$ and $-1<{\delta }_1+{\delta }_3<1,-1<{\delta }_2+{\delta }_3<1$. It is said that $(X,Y)$ has BI$q-\mathcal{G}\mathcal{E}\mathcal{V}$ distribution with parameters $s,m,\xi ,q$ and denoted by $(X,Y)\sim BIq-\mathcal{G}\mathcal{E}\mathcal{V}$, if its joint CDF (JCDF) and joint PDF (JPDF) are given by

$$F^1(x,y;s,m,q,\xi )={[1+q{\{1+\xi (-m+sx)\}}^{-(1/\xi )}]}^{-(1/q)}\times [1+({\delta }_1+{\delta }_3)\{-{[1+q{\{1+\xi (-m+sx)\}}^{-(1/\xi )}]}^{-(1/q)}+1\}+({\delta }_2+{\delta }_3)\{-{[q{\{1+\xi (-m+sy)\}}^{-(1/\xi )}+1]}^{-(1/q)}+1\}]$$

and

$$f^1(x,y;s,m,\xi ,q)=s^2{[1+\xi (-m+sx)]}^{-(1/\xi )-1}{[1+\xi (-m+sy)]}^{-\{1+(1/\xi )\}}\times {[1+q{\{1+\xi (-m+sx)\}}^{-(1/\xi )}]}^{-\{1+(1/q)\}}{[1+q{\{1+\xi (-m+sy)\}}^{-(1/\xi )}]}^{-\{1+(1/q)\}}\times [1+({\delta }_1+{\delta }_3)(1-2{(1+q{(\xi (-m+sx)+1)}^{-1/\xi })}^{-(1/q)})+({\delta }_2+{\delta }_3)\{1-2{[1+q{\{1+\xi (-m+sy)\}}^{-(1/\xi )}]}^{-(1/q)}\left\}\right]$$

where $\xi \ne 0$, and $q\ne 0$. It is said that $(X,Y)$ has BIq-Gumbel distribution and denoted by $(X,Y)\sim BIq-Gumbel$, if its JCDF and JPDF are given by

$$F^2(x,y;s,m,q)={[1+q{\mathrm{e}}^{m-sx}]}^{-(1/q)}{[1+q{\mathrm{e}}^{m-sy}]}^{-(1/q)}\times [1+({\delta }_1+{\delta }_3)\{1-{[1+q{\mathrm{e}}^{m-sx}]}^{-(1/q)}\}+({\delta }_2+{\delta }_3)\{1-{(1+q{\mathrm{e}}^{m-sy})}^{-(1/q)}\}],$$ (17)

$$f^2(x,y;s,m,q)=s^2{\mathrm{e}}^{2m-s(x+y)}{(1+q{\mathrm{e}}^{m-sx})}^{-(1/q)-1}{(1+q{\mathrm{e}}^{m-sy})}^{-\frac{1}{q}-1}\times [1+({\delta }_1+{\delta }_3)\{1-2{(1+q{\mathrm{e}}^{m-sx})}^{-(1/q)}\}+({\delta }_2+{\delta }_3)\{1-2{(1+q{\mathrm{e}}^{m-sy})}^{-(1/q)}\}],$$ (18)

where $\xi \to 0$, and $q\ne 0$.

Following the pattern adopted by the authors in [34], for $\xi \doteq 0$ and $q\ne 0$, we now derive the joint CDF (JCDF) and PDF (JPDF) of random variables X and Y, respectively, as

$$F^1(x;s,m,\xi ,q)={[1+q{\{1+\xi (-m+sx)\}}^{-(1/\xi )}]}^{-(1/q)}\times [1+({\delta }_1+{\delta }_3)\{1-{[1+q{\{1+\xi (-m+sx)\}}^{-(1/\xi )}]}^{-(1/q)}\}],$$

$$f^1(x;s,m,\xi ,q)=s{[1+\xi (-m+sx)]}^{-\{1+(1/\xi )\}}{[1+q{\{1+\xi (-m+sx)\}}^{-(1/\xi )}]}^{-\{1+(1/q)\}}\times [1+({\delta }_1+{\delta }_3)\{1-2{[1+q{\{1+\xi (-m+sx)\}}^{-1/\xi }]}^{-1/q}\}],$$

$$F^1(y;s,m,\xi ,q)={[1+q{\{1+\xi (-m+sy)\}}^{-(1/\xi )}]}^{-(1/q)}\times [1+({\delta }_2+{\delta }_3)\{1-{[1+q{(1+\xi (-m+sy))}^{-1/\xi }]}^{-(1/q)}\}],$$

$$f^1(y;s,m,\xi ,q)=s{[1+\xi (-m+sy)]}^{-\{1+(1/\xi )\}}{[1+q{\{1+\xi (-m+sx)\}}^{-(1/\xi )}]}^{-\{1+(1/q)\}}\times [1+({\delta }_2+{\delta }_3)\{1-2{[1+q{\{1+\xi (-m+sy)\}}^{-1/\xi }]}^{-1/q}\}],$$

$$F^2(x;s,m,q)=[1+q{\mathrm{e}}^{m-sx}][1+({\delta }_1+{\delta }_3)\{1-{[1+q{\mathrm{e}}^{m-sx}]}^{-(1/q)}\}],$$

$$f^2(x;s,m,\xi ,q)=s{\mathrm{e}}^{m-sx}{[1+q{\mathrm{e}}^{m-sx}]}^{-(1/q)-1}\times [1+({\delta }_1+{\delta }_3)\{1-2{\{1+q{\mathrm{e}}^{m-sx}\}}^{-(1/q)}\}],$$

$$F^2(y;s,m,q)=[1+q{\mathrm{e}}^{m-sy}][1+({\delta }_2+{\delta }_3)\{1-{[1+q{\mathrm{e}}^{m-sy}]}^{-(1/q)}\}],$$

$$f^2(y;s,m,\xi ,q)=s{\mathrm{e}}^{m-sy}{[1+q{\mathrm{e}}^{m-sy}]}^{-(1/q)-1}\times [1+({\delta }_1+{\delta }_3)\{1-2{\{1+q{\mathrm{e}}^{m-sy}\}}^{-(1/q)}\}].$$

The conditional probability density functions for $(X,Y)\sim BIq-\mathcal{G}\mathcal{E}\mathcal{V}$ are

$$f^1(y/x;s,m,q,\xi )=s{[1+\xi (-m+sy)]}^{-(1/\xi )-1}{[1+q{\{1+\xi (-m+sy)\}}^{-(1/\xi )}]}^{-(1/q)-1}\times [1+\{{\delta }_1+{\delta }_3\}[1-2{\{1+q{[1+\xi (-m+sx)]}^{-(1/\xi )}\}}^{-(1/q)}]+({\delta }_2+{\delta }_3)\{1-2{[1+q{(1+\xi (-m+sy))}^{-(1/\xi )}]}^{-(1/q)}\left\}\right]\times [1+({\delta }_1+{\delta }_3)\{1-2{(1+q{\mathrm{e}}^{m-sx})}^{-(1/q)}\left\}\right],$$

$$f^1(x/y;s,m,q,\xi )=s{[1+\xi (-m+sx)]}^{-(1/\xi )-1}{[1+q{\{1+\xi (-m+sx)\}}^{-(1/\xi )}]}^{-(1/q)-1}\times [1+\{{\delta }_1+{\delta }_3\}[1-2{\{1+q{[1+\xi (-m+sy)]}^{-(1/\xi )}\}}^{-(1/q)}]+({\delta }_2+{\delta }_3)\{1-2{[1+q{(1+\xi (-m+sx))}^{-(1/\xi )}]}^{-(1/q)}\left\}\right]\times [1+({\delta }_2+{\delta }_3)\{1-2{(1+q{\mathrm{e}}^{m-sy})}^{-(1/q)}\left\}\right],$$

$$f^2(y/x;s,m,q)=s{\mathrm{e}}^{m-sy}{[1+q{\mathrm{e}}^{m-sy}]}^{-\{1+(1/q)}[1+({\delta }_1+{\delta }_3)\{1-2{(1+q{\mathrm{e}}^{m-sx})}^{-(1/q)}\}+({\delta }_2+{\delta }_3)\{1-2{(1+q{\mathrm{e}}^{m-sy})}^{-(1/q))}\}]\times {[1+({\delta }_1+{\delta }_3)\{1-2{(1+q{\mathrm{e}}^{m-sx})}^{-(1/q)}\}]}^{-1},$$

$$f^2(x/y;s,m,\xi ,q)=s{\mathrm{e}}^{m-sy}{[1+q{\mathrm{e}}^{m-sx}]}^{-\{1+(1/q)\}}[1+({\delta }_1+{\delta }_3)\{1-2{(1+q{\mathrm{e}}^{m-sx})}^{-(1/q)}\}+({\delta }_2+{\delta }_3)\{1-2{(1+q{\mathrm{e}}^{m-sy})}^{-(1/q))}\}]\times [1+({\delta }_2+{\delta }_3)\{1-2{(1+q{\mathrm{e}}^{m-sy})}^{-(1/q)}\}].$$

The conditional moments for $(X,Y)\sim BIq-\mathcal{G}\mathcal{E}\mathcal{V}$ are

$${\mu }_{X/Y}^{1r}(y)=E(X^r|Y=y)[1+\frac{({\delta }_1+{\delta }_3)}{1+({\delta }_2+{\delta }_3)\{1-2{[1+q{(1+\xi (-m+sy))}^{-\frac{1}{\xi }}]}^{-1/q}\}}]\times \sum _{k=0}^{\infty }\sum _{l_1=0}^{\infty }\sum _{l_2=0}^{l_1}{(-1)}^{l_1-l_2}\left(\begin{array}{c}-1/q\\ k\end{array}\right)\left(\begin{array}{c}-\frac{k}{\xi }\\ l_1\end{array}\right)\left(\begin{array}{c}{l}_1\\ l_2\end{array}\right)q^k\frac{l_2{\xi }^{l_1}{m}^{{}^{l_1-l_2}}{s}^{{}^{l_2}}}{l_2+r}-\frac{2({\delta }_1+{\delta }_3)}{1+({\delta }_2+{\delta }_3)[1-2{\{1+q{(1+\xi (-m+sy))}^{-1/\xi })}^{-}(1/q)]}\times \sum _{k_1=0}^{\infty }\sum _{k_2=0}^{\infty }\sum _{l_{11}=0}^{\infty }\sum _{l_{12}=0}^{\infty }\sum _{l_{21}=0}^{l_{11}}\sum _{l_{22}=0}^{l_{12}}{(-1)}^{l_{11}+l_{12}-l_{21}-l_{22}}{\mathrm{\Omega }}_{k_1,k_2,l_{11},l_{12},l_{21},l_{22}},$$

and

$${\mu }_{Y/X}^{1r}(x)=E(Y^r|X=x)[1+\frac{({\delta }_2+{\delta }_3)}{1+({\delta }_1+{\delta }_3)\{1-2{[1+q{(1+\xi (-m+sx))}^{-1/\xi }]}^{-1/q}\}}]\times \sum _{k=0}^{\infty }\sum _{l_1=0}^{\infty }\sum _{l_2=0}^{l_1}{(-1)}^{l_1-l_2}\left(\begin{array}{c}-(1/q)\\ k\end{array}\right)\left(\begin{array}{c}-\frac{k}{\xi }\\ l_1\end{array}\right)\left(\begin{array}{c}{l}_1\\ l_2\end{array}\right)q^k\frac{l_2{\xi }^{l_1}{m}^{{}^{l_1-l_2}}{s}^{{}^{l_2}}}{l_2+r}-\frac{2({\delta }_2+{\delta }_3)}{1+({\delta }_1+{\delta }_3)[1-2{\{1+q{[1+\xi (-m+sx)]}^{-1/\xi }\}}^{-1/q}]}\times \sum _{k_1=0}^{\infty }\sum _{k_2=0}^{\infty }\sum _{l_{11}=0}^{\infty }\sum _{l_{12}=0}^{\infty }\sum _{l_{21}=0}^{l_{11}}\sum _{l_{22}=0}^{l_{12}}{(-1)}^{l_{11}+l_{12}-l_{21}-l_{22}}{\mathrm{\Omega }}_{k_1,k_2,l_{11},l_{12},l_{21},l_{22}},$$

and for $(X,Y)\sim q-Gumbel$ are

$${\mu }_{X/Y}^{2r}(y)=E(X^r|Y=y)[1+\frac{{\delta }_1+{\delta }_3}{1+({\delta }_2+{\delta }_3)\{1-2[1-2{(1+q{\mathrm{e}}^{m-sy})}^{-1/q}]\}}]\sum _{k=0}^{\infty }\sum _{l_1=0}^{\infty }\sum _{l_2=0}^{l_1}{(-1)}^{l_2}\left(\begin{array}{c}-\frac{1}{q}\\ k\end{array}\right)q^k\frac{l_2{k}^{l_1}{m}^{{}^{l_1-l_2}}{s}^{{}^{l_2}}}{(l_2+r).l_1!}-\frac{2({\delta }_1+{\delta }_3)}{1+({\delta }_2+{\delta }_3)(1-2(1-2{(1+q{e}^{m-sx})}^{-\frac{1}{q}}))}\sum _{k_1=0}^{\infty }\sum _{k_2=0}^{\infty }\sum _{l_{11}=0}^{\infty }\sum _{l_{12}=0}^{\infty }\sum _{l_{21}=0}^{l_{11}}\sum _{l_{22}=0}^{l_{12}}{(-1)}^{l_{21}+l_{22}}{l}_{22}\times \left(\begin{array}{c}-\frac{1}{q}\\ k_1\end{array}\right)\left(\begin{array}{c}-\frac{1}{q}\\ k_2\end{array}\right)\frac{k_1^{l_{11}}{k}_2^{l_{12}}{m}^{{}^{l_{11}+l_{12}-l_{21}-l_{22}}}{s}^{{}^{l_{21}+l_{22}}}}{l_{11}!l_{12}!},$$

$${\mu }_{Y/X}^{2r}(x)=E(Y^r|X=x)[1+\frac{{\delta }_2+{\delta }_3}{1+({\delta }_1+{\delta }_3)\{1-2[1-2{(1+q{\mathrm{e}}^{m-sx})}^{-\frac{1}{q}}]\}}]\sum _{k=0}^{\infty }\sum _{l_1=0}^{\infty }\sum _{l_2=0}^{l_1}{(-1)}^{l_2}\left(\begin{array}{c}-\frac{1}{q}\\ k\end{array}\right)q^k\frac{l_2{k}^{l_1}{m}^{{}^{l_1-l_2}}{s}^{{}^{l_2}}}{(l_2+r).l_1!}-\frac{2({\delta }_1+{\delta }_3)}{1+({\delta }_2+{\delta }_3)(1-2(1-2{(1+q{e}^{m-sy})}^{-\frac{1}{q}}))}\sum _{k_1=0}^{\infty }\sum _{k_2=0}^{\infty }\sum _{l_{11}=0}^{\infty }\sum _{l_{12}=0}^{\infty }\sum _{l_{21}=0}^{l_{11}}\sum _{l_{22}=0}^{l_{12}}{(-1)}^{l_{21}+l_{22}}{l}_{22}\times \left(\begin{array}{c}-\frac{1}{q}\\ k_1\end{array}\right)\left(\begin{array}{c}-\frac{1}{q}\\ k_2\end{array}\right)\frac{k_1^{l_{11}}{k}_2^{l_{12}}{m}^{{}^{l_{11}+l_{12}-l_{21}-l_{22}}}{s}^{{}^{l_{21}+l_{22}}}}{l_{11}!l_{12}!},$$

The joint moment for $(X,Y)\sim BIq-GEV$ ${\mu }_{r,v}=E(X^r{Y}^v)=(1+{\delta }_1+{\delta }_2+2{\delta }_3)\sum _{k_1=0}^{\infty }\sum _{k_2=0}^{\infty }\sum _{l_{11}=0}^{\infty }\sum _{l_{12}=0}^{\infty }\sum _{l_{21}=0}^{l_{11}}\sum _{l_{22}=0}^{l_{12}}\{\frac{{(-1)}^{l_{11}+l_{12}-l_{21}-l_{22}}}{(l_{21}+r)(l_{22}+v)}$

$\times {\mathrm{\Omega }}_{k_1,k_2,l_{11},l_{12},l_{21},l_{22}}\}-2(({\delta }_1+{\delta }_3)\sum _{k=0}^{\infty }\sum _{l_1=0}^{\infty }\sum _{l_2=0}^{l_1}{(-1)}^{l_1-l_2}\left(\begin{array}{c}-\frac{1}{q}\\ k\end{array}\right)\left(\begin{array}{c}-\frac{k}{\xi }\\ l_1\end{array}\right)\left(\begin{array}{c}{l}_1\\ l_2\end{array}\right)q^k\frac{l_2{\xi }^{l_1}{m}^{{}^{l_1-l_2}}{s}^{{}^{l_2}}}{l_2+v}$

$+({\delta }_2+{\delta }_3)\sum _{k=0}^{\infty }\sum _{l_1=0}^{\infty }\sum _{l_2=0}^{l_1}{(-1)}^{l_1-l_2}\left(\begin{array}{c}-\frac{1}{q}\\ k\end{array}\right)\left(\begin{array}{c}-\frac{k}{\xi }\\ l_1\end{array}\right)\left(\begin{array}{c}{l}_1\\ l_2\end{array}\right)q^k\frac{l_2{\xi }^{l_1}{m}^{{}^{l_1-l_2}}{s}^{{}^{l_2}}}{l_2+r})$

$\times \sum _{k_1=0}^{\infty }\sum _{k_2=0}^{\infty }\sum _{l_{11}=0}^{\infty }\sum _{l_{12}=0}^{\infty }\sum _{l_{21}=0}^{l_{11}}\sum _{l_{22}=0}^{l_{12}}{(-1)}^{l_{11}+l_{12}-l_{21}-l_{22}}{\mathrm{\Omega }}_{k_1,k_2,l_{11},l_{12},l_{21},l_{22}}$,

where, ${\mathrm{\Omega }}_{k_1,k_2,l_{11},l_{12},l_{21},l_{22}}=\left(\begin{array}{c}-\frac{1}{q}\\ k_1\end{array}\right)\left(\begin{array}{c}-\frac{1}{q}\\ k_2\end{array}\right)\left(\begin{array}{c}-\frac{k_1}{\xi }\\ l_{11}\end{array}\right)\left(\begin{array}{c}-\frac{k_2}{\xi }\\ l_{12}\end{array}\right)\left(\begin{array}{c}{l}_{11}\\ l_{21}\end{array}\right)\left(\begin{array}{c}{l}_{12}\\ l_{22}\end{array}\right)\times q^{k_1+k_2}{\xi }^{l_{11}+l_{12}}{l}_{22}{m}^{{}^{l_{11}+l_{12}-l_{21}-l_{22}}}{s}^{{}^{l_{21}+l_{22}}}$.

The bivariate reliability function for $(X,Y)\sim BIq-\mathcal{G}\mathcal{E}\mathcal{V}$ is

$$R^1(x,y)=1-{[1+q{\{\xi (-m+sx)+1\}}^{-\frac{1}{\xi }}]}^{-\frac{1}{q}}[1+({\delta }_1+{\delta }_3)\{1-{[1+q{(1+\xi (-m+sx))}^{-\frac{1}{\xi }}]}^{-\frac{1}{q}}\}]-{[1+q{\{1+\xi (-m+sy)\}}^{-\frac{1}{\xi }}]}^{-\frac{1}{q}}[1+({\delta }_2+{\delta }_3)(1-{\{1+q{(1+\xi (-m+sy))}^{-\frac{1}{\xi }}\}}^{-\frac{1}{q}}\left)\right]+{[1+q{(\xi (-m+sx)+1)}^{-\frac{1}{\xi }}]}^{-\frac{1}{q}}{[1+q{\{1+\xi (-m+sy)\}}^{-\frac{1}{\xi }}]}^{-\frac{1}{q}}\times [1+({\delta }_1+{\delta }_3)\{1-{[1+q{\{1+\xi (-m+sx)\}}^{-\frac{1}{\xi }}]}^{-\frac{1}{q}}\}+({\delta }_2+{\delta }_3)\times \{1-{[1+q{\{1+\xi (-m+sy)\}}^{-\frac{1}{\xi }}]}^{-\frac{1}{q}}\}],$$ (19)

and for $(X,Y)\sim q-Gumbel$ is

$$R^2(x,y)=1-(1+q{\mathrm{e}}^{m-sx})[1+({\delta }_1+{\delta }_3)\{1-{(1+q{\mathrm{e}}^{m-sx})}^{-(1/q)}\}]-(1+q{\mathrm{e}}^{m-sy})\times [1+({\delta }_2+{\delta }_3)\{1-{(1+q{\mathrm{e}}^{m-sy})}^{-(1/q)}\}]+s^2{\mathrm{e}}^{2m-s(x+y)}{(1+q{\mathrm{e}}^{m-sx})}^{-\{1+(1/q)\}}\times {(1+q{\mathrm{e}}^{m-sy})}^{-\{1+(1/q)\}}[1+({\delta }_1+{\delta }_3)\{1-2{(1+q{\mathrm{e}}^{m-sx})}^{-\frac{1}{q}}\}+({\delta }_2+{\delta }_3)\{1-2{(1+q{\mathrm{e}}^{m-sy})}^{-(1/q)}\}].$$

The bivariate hazard rate function for $(X,Y)\sim BIq-\mathcal{G}\mathcal{E}\mathcal{V}$, is given by

$$h^1(x,y)=[s^2{[1+\xi (-m+sx)]}^{-\{1+(1/q)\}}{[1+\xi (-m+sy)]}^{-\{1+(1/q)\}}\times {[1+q{\{1+\xi (-m+sx)\}}^{-\frac{1}{\xi }}]}^{-\{1+(1/q)\}}{[1+q{\{1+\xi (-m+sy)\}}^{-\frac{1}{\xi }}]}^{-\{1+(1/q)\}}\times \{1+({\delta }_1+{\delta }_3)[1-2{\{1+q{[1+\xi (-m+sx)]}^{-(1/\xi )}\}}^{-(1/q)}]+({\delta }_2+{\delta }_3)[1-2{\{1+q{[1+\xi (-m+sy)]}^{-(1/\xi )}\}}^{-(1/q)}]\left\}\right]1-{(1+q{(\xi (-m+sx)+1)}^{-\frac{1}{\xi }})}^{-(1/q)}[1+({\delta }_1+{\delta }_3)(1-{(1+q{(\xi (-m+sx)+1)}^{-\frac{1}{\xi }})}^{-\frac{1}{q}})]-{(1+q{(\xi (-m+sy)+1)}^{-\frac{1}{\xi }})}^{-\frac{1}{q}}[1+({\delta }_2+{\delta }_3)(1-{(1+q{(\xi (-m+sy)+1)}^{-\frac{1}{\xi }})}^{-\frac{1}{q}})]+{(1+q{(\xi (-m+sx)+1)}^{-\frac{1}{\xi }})}^{-\frac{1}{q}}{(1+q{(\xi (-m+sy)+1)}^{-\frac{1}{\xi }})}^{-\frac{1}{q}}\times [1+({\delta }_1+{\delta }_3)\{1-{(1+q{[1+\xi (-m+sx)]}^{-(1/\xi )})}^{-\frac{1}{q}}\}{+({\delta }_2+{\delta }_3)\{1-{[1+q{\{1+\xi (-m+sy)\}}^{-(1/\xi )}]}^{-(1/\xi )}\}]}^{-1},$$ (20)

and for $(X,Y)\sim q-Gumbel$ is

$$h^2(x,y)=[s^2{\mathrm{e}}^{2m-s(x+y)}{(1+q{\mathrm{e}}^{m-sx})}^{-(1/q)-1}{(1+q{\mathrm{e}}^{m-sy})}^{-(1/q)-1}\times [1+({\delta }_1+{\delta }_3)\{1-2{(1+q{\mathrm{e}}^{m-sx})}^{-(1/q)}\}+({\delta }_2+{\delta }_3)\{1-2{(1+q{\mathrm{e}}^{m-sy})}^{-(1/q)}\}\left]\right]\times [1-(1+q{\mathrm{e}}^{m-sx})\{1+({\delta }_1+{\delta }_3)[1-{(1+q{\mathrm{e}}^{m-sx})}^{-(1/q)}]\}-(1+q{\mathrm{e}}^{m-sy})\{1+({\delta }_2+{\delta }_3)(1-{(1+q{\mathrm{e}}^{m-sy})}^{-(1/q)})\}+s^2{\mathrm{e}}^{2m-s(x+y)}{(1+q{\mathrm{e}}^{m-sx})}^{-\frac{1}{q}-1}{(1+q{\mathrm{e}}^{m-sy})}^{-(1/q)-1}{\times [1+({\delta }_1+{\delta }_3)(1-2{(1+q{\mathrm{e}}^{m-sx})}^{-(1/q)})+({\delta }_2+{\delta }_3)(1-2{(1+q{\mathrm{e}}^{m-sy})}^{-(1/q)})]]}^{-1}.$$

The copula function for $(X,Y)\sim BIq-\mathcal{G}\mathcal{E}\mathcal{V}$ is given by

$$c^1(u,v)=[1+({\delta }_1+{\delta }_3)\{1-2{[1+q{(1+\xi (-m+sx))}^{-(1/\xi )}]}^{-(1/q)}\}+({\delta }_2+{\delta }_3)\{1-2{[1+q{\{1+\xi (-m+sy)\}}^{-(1/\xi )}]}^{-(1/q)}\}]\times {[1+({\delta }_1+{\delta }_3)\{1-2{(1+q{(1+\xi (-m+sx))}^{-(1/\xi )})}^{-(1/q)}\}]}^{-1}\times {[1+({\delta }_2+{\delta }_3)\{1-2{[1+q{(1+\xi (-m+sy))}^{-(1/\xi )}]}^{-(1/q)}\}]}^{-1},$$

and for $(X,Y)\sim q-Gumbel$ is

$$c^2(u,v)=[1+({\delta }_1+{\delta }_3)(1-2{(1+q{\mathrm{e}}^{m-sx})}^{-(1/q)})+({\delta }_2+{\delta }_3)(1-2{(1+q{\mathrm{e}}^{m-sy})}^{-(1/q)})]\times {[1+({\delta }_1+{\delta }_3)(1-2{(1+q{\mathrm{e}}^{m-sx})}^{-(1/q)})]}^{-1}{[1+({\delta }_2+{\delta }_3)(1-2{(1+q{\mathrm{e}}^{m-sy})}^{-(2/q)})]}^{-1}.$$

3.2. Estimation

The principal of maximum likelihood, a statistical technique to optimize the probability of diagnosing the provided sample data, entails determining which of the model parameter values are most likely to match the observed data. In this regard, a random sample of stochastic variables $(x_1,y_1),(x_2,y_2)...(x_n,y_n)$ is taken independently as $(X,Y)\sim BIq-\mathcal{G}\mathcal{E}\mathcal{V}$, assumed to be associated positively to the distribution. Consequently, the maximum log-likelihood function $L^1(s,m,\xi ,q)$ is given as

$$2n\ln s-\{1+(1/\xi )\}\sum _{i=1}^n[\ln \{1+(-m+s{x}_i)\xi \}+\ln \{1+(-m+s{y}_i)]xi\}]-\{1+(1/q)\}\times \sum _{i=1}^n\{\ln [1+q{(1+\xi (-m+s{x}_i))}^{-1/\xi }]+\ln [1+q{\{1+\xi (-m+s{y}_i)\}}^{-1/\xi }]\}+\sum _{i=1}^n[\ln \{1+({\delta }_1+{\delta }_3)[1-2{\{1+q{[1+\xi (-m+s{y}_i)]}^{-1/\xi }\}}^{-1/q}]+({\delta }_2+{\delta }_3)[1-2{\{1+q{[1+\xi (-m+s{y}_i)]}^{-1/\xi }\}}^{-1/q}\left]\right\}]$$

By computing the following system of solutions computationally, one can establish the maximum likelihood estimators of quantities.

$$\begin{gathered} L_s^1=\frac{2n}{s}-(\xi +1)\sum _{i=1}^n[\frac{x_i}{1+\xi (-m+s{x}_i)}+\frac{y_i}{1+\xi (-m+s{y}_i)}]+(q+1)\sum _{i=1}^n[\frac{x_i{\{1+\xi (-m+s{x}_i)\}}^{-\frac{1}{\xi }-1}}{1+q{\{1+\xi (-m+s{x}_i)\}}^{-(1/\xi )}}+\frac{y_i{\{1+\xi (-m+s{y}_i)\}}^{-\frac{1}{\xi }-1}}{1+q{\{1+\xi (-m+s{y}_i)\}}^{-(1/\xi )}}]-2({\delta }_1+{\delta }_3)\sum _{i=1}^n[{\{1+\xi (-m+s{x}_i)\}}^{-(1/\xi )-1}{\{1+q{[\xi (-m+s{x}_i)+1]}^{-1/\xi }\}}^{-(1/q)-1}{x}_i \\ \times \{1+({\delta }_1+{\delta }_3)[1-2{\{1+q{[1+\xi (-m+s{x}_i)]}^{-\frac{1}{\xi }}\}}^{-1/q}]+({\delta }_2+{\delta }_3){\times [1-2{\{1+q{[1+\xi (-m+s{y}_i)]}^{-1/\xi }\}}^{-1/q}]\}}^{-1}]-2({\delta }_2+{\delta }_3)\times \sum _{i=1}^n[{\{1+\xi (-m+s{y}_i)\}}^{-\frac{1}{\xi }-1}{\{1+q{[1+\xi (-m+s{y}_i)]}^{-1/\xi }\}}^{-\frac{1}{q}-1}{x}_i \\ \times \{1+({\delta }_1+{\delta }_3)[1-2{\{1+q{[1+\xi (-m+s{x}_i)]}^{-1/\xi }\}}^{-1/q}]{+({\delta }_2+{\delta }_3)[1-2{\{1+q{[1+\xi (-m+s{y}_i)]}^{-1/\xi }\}}^{-1/q}]\}}^{-1}], \end{gathered}$$

$$\begin{gathered} L_m^1=(\xi +1)\sum _{i=1}^n\{\frac{1}{1+\xi (-m+s{x}_i)}+\frac{1}{1+\xi (-m+s{y}_i)}\}+(1+\frac{1}{q})\sum _{i=1}^n[\frac{1+\xi {(-m+s{x}_i)}^{-\frac{1}{\xi }-1}}{1+q{\{1+\xi (-m+s{x}_i)\}}^{-1/\xi }}+\frac{{\{1+\xi (-m+s{y}_i)\}}^{-\frac{1}{\xi }-1}}{1+q{\{1+\xi (-m+s{y}_i)\}}^{1/\xi }}]+2({\delta }_1+{\delta }_3)\sum _{i=1}^n\{{[1+\xi (-m+s{x}_i)]}^{-\frac{1}{\xi }-1}{[1+q{\{1+\xi (-m+s{x}_i)\}}^{-1/\xi }]}^{-\frac{1}{q}-1} \\ \times [1+({\delta }_1+{\delta }_3)[1-2{\{1+q{[1+\xi (-m+s{x}_i)]}^{-1/\xi }\}}^{-1/q}]{+({\delta }_2+{\delta }_3)\{1-2{[1+q{\{1+\xi (-m+s{y}_i)\}}^{-1/\xi }]}^{-1/q}\}]}^{-1}\}+2({\delta }_2+{\delta }_3)\sum _{i=1}^n\{{[1+\xi (-m+s{y}_i)]}^{-\frac{1}{\xi }-1}{[1+q{\{1+\xi (-m+s{y}_i)\}}^{-1/\xi }]}^{-\frac{1}{q}-1} \\ \times [1+({\delta }_1+{\delta }_3)\{1-2{[1+q{\{1+\xi (-m+s{x}_i)\}}^{-1/\xi }]}^{-1/q}\}{+({\delta }_2+{\delta }_3)\{1-2{[1+q{\{1+\xi (-m+s{y}_i)\}}^{-1/\xi }]}^{-1/q}\}]}^{-1}\}, \end{gathered}$$

$$\begin{gathered} L_{\xi }^1=-(1+\frac{1}{\xi })\sum _{i=1}^n[\frac{-m+s{x}_i}{1+\xi (-m+s{x}_i)}+\frac{-m+s{y}_i}{1+\xi (-m+s{y}_i)}]+{\xi }^{-2}\sum _{i=1}^n\ln \{1+\xi (-m+s{x}_i)\}+{\xi }^{-2}\sum _{i=1}^n\ln \{1+\xi (-m+s{y}_i)\}+(q+1){\xi }^{-2}\sum _{i=1}^n\frac{1+\xi {[-m+s{x}_i]}^{-1/\xi }[\ln (\xi (-m+s{x}_i)+1)-\frac{\xi (-m+s{x}_i)}{\xi (-m+s{x}_i)+(q+1){\xi }^{-2}}]}{1+q{[\xi (-m+s{x}_i)+1]}^{-1/\xi }} \\ +(q+1){\xi }^{-2}\sum _{i=1}^n\frac{{[1+\xi (-m+s{y}_i)]}^{-1/\xi }[\ln \{1+\xi (-m+s{y}_i)\}-\frac{\xi (-m+s{y}_i)}{1+\xi (-m+s{y}_i)}]}{1+q{\{1+\xi (-m+s{y}_i)\}}^{-1/\xi }} \\ +2({\delta }_1+{\delta }_3)\sum _{i=1}^n\{{[1+\xi (-m+s{x}_i)]}^{-1/\xi }[\ln (\xi (-m+s{x}_i)+1)-\frac{\xi (-m+s{x}_i)}{\xi (-m+s{x}_i)+1}]\times {[1+q{\{1+\xi (-m+s{x}_i)\}}^{-1/xi}]}^{-(1/q)-1}[1+({\delta }_1+{\delta }_3)\{1-2{[1+q{\{1+\xi (-m+s{x}_i)\}}^{-1/\xi }]}^{-1/q}\}{+({\delta }_2+{\delta }_3)(1-2{[1+q{\{1+\xi (-m+s{y}_i)\}}^{-1/\xi }]}^{-1/q})]}^{-1}\}+2({\delta }_2+{\delta }_3)\sum _{i=1}^n\{{[1+\xi (-m+s{y}_i)]}^{-1/\xi }\{\ln [1+\xi (-m+s{y}_i)]-\frac{-m+\xi (s{y}_i)}{1+\xi (-m+s{y}_i)}\}\times {[1+q{\{1+\xi (-m+s{y}_i)\}}^{-1/\xi }]}^{-(1/q)-1}[1+({\delta }_1+{\delta }_3)\{1-2{[1+q{\{1+\xi (-m+s{x}_i)\}}^{-\frac{1}{\xi }}]}^{-\frac{1}{q}}\}{+({\delta }_2+{\delta }_3)(1-2{[1+q{\{1+\xi (-m+s{y}_i)\}}^{-1/\xi }]}^{-1/q})]}^{-1}\} \end{gathered}$$

$$L_q^1=-(\frac{1}{q}+1)\sum _{i=1}^n\{\frac{{(1+\xi (-m+s{x}_i))}^{-1/\xi }}{1+q{(1+\xi (-m+s{x}_i))}^{-1/\xi }}+\frac{{(1+\xi (-m+s{y}_i))}^{-1/\xi }}{1+q{(1+\xi (-m+s{y}_i))}^{-1/\xi }}\}+\frac{1}{q^2}\sum _{i=1}^n\{\ln (1+q{(1+\xi (-m+s{x}_i))}^{-1/\xi })+\ln (1+q{(1+\xi (-m+s{y}_i))}^{-1/\xi })\}-\frac{2}{q^2}({\delta }_1+{\delta }_3)\sum _{i=1}^n\{[\ln (1+q{(1+\xi (-m+s{x}_i))}^{-1/\xi })-\frac{q{(1+\xi (-m+s{x}_i))}^{-1/\xi }}{1+q{(1+\xi (-m+s{x}_i))}^{-1/\xi }}]\times {(1+q{(1+\xi (-m+s{x}_i))}^{-1/\xi })}^{-1/q}(1+({\delta }_1+{\delta }_3)(1-2{(1+q{(1+\xi (-m+s{x}_i))}^{-1/\xi })}^{-1/q})+{({\delta }_2+{\delta }_3)(1-2{(1+q{(1+\xi (-m+s{y}_i))}^{-1/\xi })}^{-1/q}))}^{-1}\}+2({\delta }_2+{\delta }_3)\sum _{i=1}^n\{[\ln (1+q{(1+\xi (-m+s{y}_i))}^{-1/\xi })-\frac{q{(1+\xi (-m+s{y}_i))}^{-1/\xi }}{1+q{(1+\xi (-m+s{y}_i))}^{-1/\xi }}]\times {(1+q{(1+\xi (-m+s{y}_i))}^{-1/\xi })}^{-1/q}(1+({\delta }_1+{\delta }_3)(1-2{(1+q{(1+\xi (-m+s{x}_i))}^{-1/\xi })}^{-1/q})+{({\delta }_2+{\delta }_3)(1-2{(1+q{(1+\xi (-m+s{y}_i))}^{-1/\xi })}^{-1/q}))}^{-1}\}$$

$$\begin{gathered} L_{{\delta }_1}^1=\sum _{i=1}^n\{1-2{[1+q{\{1+\xi (-m+s{x}_i)\}}^{-1/\xi }]}^{-1/q} \\ \times [1+({\delta }_1+{\delta }_3)\{1-2{[1+q{[1+\xi (-m+s{x}_i)]}^{-1/\xi }]}^{-1/q}\}+{({\delta }_2+{\delta }_3)\{1-2{[1+q{\{1+\xi (-m+s{y}_i)\}}^{-1/\xi }]}^{-1/q}\}]}^{-1}\}, \end{gathered}$$

$$\begin{gathered} L_{{\delta }_2}^1=\sum _{i=1}^n\{1-2{[q{\{1+\xi (-m+s{y}_i)\}}^{-1/\xi }+1]}^{-1/q} \\ \times [1+({\delta }_1+{\delta }_3)\{1-2{[1+q{\{1+\xi (-m+s{x}_i)\}}^{-1/\xi }]}^{-1/q}\}+{({\delta }_2+{\delta }_3)\{1-2{[1+q{\{1+\xi (-m+s{y}_i)\}}^{-1/\xi }]}^{-1/q}\}]}^{-1}\}, \end{gathered}$$

$$\begin{gathered} L_{{\delta }_3}^1=2\sum _{i=1}^n\{1-{[1+q{\{1+\xi (-m+s{x}_i)\}}^{-1/\xi }]}^{-1/q}-{[1+q{\{1+\xi (-m+s{y}_i)\}}^{-1/\xi }]}^{-1/q} \\ \times [1+({\delta }_1+{\delta }_3)\{-2{[1+q{(1+\xi (-m+s{x}_i))}^{-1/\xi }]}^{-1/q}+1\}+{({\delta }_2+{\delta }_3)\{-2{[1+q{\{1+\xi (-m+s{y}_i)\}}^{-1/\xi }]}^{-1/q}+1\}]}^{-1}\}. \end{gathered}$$

For a random sample $(x_1,y_1),(x_2,y_2)...(x_n,y_n)$, taken independently from the random variable $(X,Y)\sim q-Gumbel$, the maximum log-likelihood function is

$$\begin{gathered} L^2(s,m,q)=2n\ln s+\sum _{i=1}^n(-s{x}_i+m)+\sum _{i=1}^n(-s{y}_i+m)-(\frac{1}{q}+1)\sum _{i=1}^n\{\ln (1+q{\mathrm{e}}^{-s{x}_i+m})+\ln (1+q{\mathrm{e}}^{-s{y}_i+m}))\}+\sum _{i=1}^n\{\ln [1+({\delta }_1+{\delta }_3)(1-2{(1+q{\mathrm{e}}^{-s{x}_i+m})}^{-\frac{1}{q}}) \\ +({\delta }_2+{\delta }_3)(1-2{(1+q{\mathrm{e}}^{-s{y}_i+m})}^{-\frac{1}{q}})]\}. \end{gathered}$$

The maximum likelihood estimators of parameters can be obtained by numerically solving the following system of non-linear equations.

$$\begin{gathered} L_s^2=\frac{2n}{s}-\sum _{i=1}^n(x_i+y_i)+q(\frac{1}{q}+1)\sum _{i=1}^n(\frac{x_i{\mathrm{e}}^{-s{x}_i+m}}{1+q{\mathrm{e}}^{-s{x}_i+m}}+\frac{y_i{\mathrm{e}}^{-s{y}_i+m}}{1+q{\mathrm{e}}^{-s{y}_i+m}})-2({\delta }_1+{\delta }_3)\sum _{i=1}^n\frac{x_i{\mathrm{e}}^{-s{x}_i+m}(1-2{(1+q{\mathrm{e}}^{-s{y}_i+m})}^{-\frac{1}{q}-1})}{1+({\delta }_1+{\delta }_3)(1-2{(1+q{\mathrm{e}}^{-s{x}_i+m})}^{-\frac{1}{q}})+({\delta }_2+{\delta }_3)(1-2{(1+q{\mathrm{e}}^{-s{y}_i+m})}^{-\frac{1}{q}})} \\ -2({\delta }_2+{\delta }_3)\sum _{i=1}^n\frac{y_i{\mathrm{e}}^{-s{y}_i+m}(1-2{(1+q{\mathrm{e}}^{-s{y}_i+m})}^{-\frac{1}{q}-1})}{1+({\delta }_1+{\delta }_3)(1-2{(1+q{\mathrm{e}}^{-s{x}_i+m})}^{-\frac{1}{q}})+({\delta }_2+{\delta }_3)(1-2{(1+q{\mathrm{e}}^{-s{y}_i+m})}^{-\frac{1}{q}})}, \end{gathered}$$

$$L_m^2=2n-q(\frac{1}{q}+1)\sum _{i=1}^n(\frac{{\mathrm{e}}^{-s{x}_i+m}}{1+q{\mathrm{e}}^{-s{x}_i+m}}+\frac{{\mathrm{e}}^{-s{y}_i+m}}{1+q{\mathrm{e}}^{-s{y}_i+m}})+2({\delta }_1+{\delta }_3)\sum _{i=1}^n\frac{{\mathrm{e}}^{-s{x}_i+m}(1-2{(1+q{\mathrm{e}}^{-s{x}_i+m})}^{-\frac{1}{q}-1})}{1+({\delta }_1+{\delta }_3)(1-2{(1+q{\mathrm{e}}^{-s{x}_i+m})}^{-\frac{1}{q}})+({\delta }_2+{\delta }_3)(1-2{(1+q{\mathrm{e}}^{-s{y}_i+m})}^{-\frac{1}{q}})}+2({\delta }_2+{\delta }_3)\sum _{i=1}^n\frac{{\mathrm{e}}^{-s{y}_i+m}(1-2{(1+q{\mathrm{e}}^{-s{y}_i+m})}^{-\frac{1}{q}-1})}{1+({\delta }_1+{\delta }_3)(1-2{(1+q{\mathrm{e}}^{-s{x}_i+m})}^{-\frac{1}{q}})+({\delta }_2+{\delta }_3)(1-2{(1+q{\mathrm{e}}^{-s{y}_i+m})}^{-\frac{1}{q}})},$$

$$\begin{gathered} L_q^2=\frac{1}{q^2}\sum _{i=1}^n\{\ln (1+q{\mathrm{e}}^{-s{x}_i+m})+\ln (1+q{\mathrm{e}}^{-s{y}_i+m})\}-(\frac{1}{q}+1)\sum _{i=1}^n(\frac{{\mathrm{e}}^{-s{x}_i+m}}{1+q{\mathrm{e}}^{-s{x}_i+m}}+\frac{{\mathrm{e}}^{-s{y}_i+m}}{1+q{\mathrm{e}}^{-s{y}_i+m}})-2({\delta }_1+{\delta }_3)\sum _{i=1}^n\frac{{(1+q{e}^{m-s{x}_i})}^{-\frac{1}{q}}(\frac{1}{q^2}\ln (2(1+q{\mathrm{e}}^{-s{x}_i+m}))-\frac{1}{2q}{e}^{m-s{x}_i}-{(1+q{\mathrm{e}}^{-s{x}_i+m})}^{-1})}{1+({\delta }_1+{\delta }_3)(1-2{(1+q{\mathrm{e}}^{-s{x}_i+m})}^{-\frac{1}{q}})+({\delta }_2+{\delta }_3)(1-2{(1+q{\mathrm{e}}^{-s{y}_i+m})}^{-\frac{1}{q}})} \\ -2({\delta }_2+{\delta }_3)\sum _{i=1}^n\frac{{(1+q{\mathrm{e}}^{-s{y}_i+m})}^{-\frac{1}{q}}(\frac{1}{q^2}\ln (2(1+q{\mathrm{e}}^{-s{y}_i+m}))-\frac{1}{2q}{\mathrm{e}}^{-s{y}_i+m}-{(1+q{\mathrm{e}}^{-s{y}_i+m})}^{-1})}{1+({\delta }_1+{\delta }_3)(1-2{(1+q{\mathrm{e}}^{-s{x}_i+m})}^{-\frac{1}{q}})+({\delta }_2+{\delta }_3)(1-2{(1+q{\mathrm{e}}^{-s{y}_i+m})}^{-\frac{1}{q}})} \end{gathered}$$

$$L_{{\delta }_1}^2=\sum _{i=1}^n\frac{(1-2{(1+q{\mathrm{e}}^{-s{x}_i+m})}^{-\frac{1}{q}})}{1+({\delta }_1+{\delta }_3)(1-2{(1+q{\mathrm{e}}^{-s{x}_i+m})}^{-\frac{1}{q}})+({\delta }_2+{\delta }_3)(1-2{(1+q{\mathrm{e}}^{-s{y}_i+m})}^{-\frac{1}{q}})},$$

$$L_{{\delta }_2}^2=\sum _{i=1}^n\frac{(1-2{(1+q{\mathrm{e}}^{-s{y}_i+m})}^{-\frac{1}{q}})}{1+({\delta }_1+{\delta }_3)(1-2{(1+q{\mathrm{e}}^{-s{x}_i+m})}^{-\frac{1}{q}})+({\delta }_2+{\delta }_3)(1-2{(1+q{\mathrm{e}}^{-s{y}_i+m})}^{-\frac{1}{q}})},$$

$$L_{{\delta }_3}^1=2\sum _{i=1}^n\frac{(1-2{(1+q{\mathrm{e}}^{-s{x}_i+m})}^{-\frac{1}{q}})+[1-2{(q{\mathrm{e}}^{-s{y}_i+m}+1)}^{-\frac{1}{q}}]}{1+({\delta }_1+{\delta }_3)(1-2{(1+q{\mathrm{e}}^{-s{x}_i+m})}^{-\frac{1}{q}})+({\delta }_2+{\delta }_3)(1-2{(1+q{\mathrm{e}}^{-s{y}_i+m})}^{-\frac{1}{q}})}.$$

3.3. Simulation

Here statistical properties of the constructed bi-variate variables $(X,Y)\sim BIq-\mathcal{G}\mathcal{E}\mathcal{V}$ and $(X,Y)\sim q-Gumbel$ are scrutinized whenever the parameters' values vary. Four Cases (case 1 to case 4) are allocated to $BIq-\mathcal{G}\mathcal{E}\mathcal{V}$ and the case 5 for the $q-Gumbel$.

Case 1: Consider Figure 8, Figure 9, Figure 10 for the parameters $(s,m,\xi ,q,{\delta }_1,{\delta }_2,{\delta }_3)=(0.7,0.2,0.5,0.3,0.2,-0.4,0.6)$ keenly. The JPDF (8-a) is defined for $(x,y)\ge (0,0)$ and its JCDF (8-b) approaches 1 whenever $(x,y)\to (11,11)$. The visualizations (9:c-e) signify the bivariate hazard function as a decreasing function which approaches to zero with cantor plot in sighting. The high correlation is easily observed in the interval $(u,v)\in (0,0.5)$. In Fig. 10(f-i), one can rightly observe right skewness pattern for the marginal PDFs and CDFs approach 1 as $(x,y)\to (11,11)$. Immediately, we can notice that the statistical quantities of this case supported using this model for ill-conditioned positive right skewed data.

Figure 8.

JPDF (8-f; 8-h) and JCDF (8-g; 8-i) of $BIq-\mathcal{G}\mathcal{E}\mathcal{V}$ distribution for Case-1.

Figure 9.

Bivariate hazard function of $BIq-\mathcal{G}\mathcal{E}\mathcal{V}$ distribution for Case-1.

Figure 10.

Marginal PDFs and CDFs of $BIq-\mathcal{G}\mathcal{E}\mathcal{V}$ distribution for Case-1.

Case 2: For the parameters $(s,m,\xi ,q,{\delta }_1,{\delta }_2,{\delta }_3)=(0.6,1.2,-0.4,0.8,0.6,0.4,0.3)$, Figure 11, Figure 12, Figure 13 are considered. The different shapes of the JPDF (11-a) in relation to JCDF (11-b) are defined for $(x,y)⪅(3,3)$. The JCDF approaches 1 whenever $(x,y)\to (6,6)$. The approximately constant bivariate hazard function along with correlation structure for $(u,v)\in (0,2)$ is displayed in Fig. 12(c-e). This characterizes the failure probability of not much change over time, as long as the factors are linked. The marginal PDFs (13-f; 13-h) as left skewed with marginal CDFs (13-g; 13-i) approach 1 for $(x,y)\to (6,6)$ are graphically characterised as well. This attributes the model, in this case, suggestive to its applicability for negative tailed data.

Figure 11.

Joint density & distribution function of $BIq-\mathcal{G}\mathcal{E}\mathcal{V}$ distribution for Case-2.

Figure 12.

Plots of failure rate with correlation structure for Case-2 of $BIq-\mathcal{G}\mathcal{E}\mathcal{V}$ distribution.

Figure 13.

Marginal PDFs and CDFs of $BIq-\mathcal{G}\mathcal{E}\mathcal{V}$ distribution for Case-2.

Case 3: In Fig. 14, consider the parameters $(s,m,\xi ,q,{\delta }_1,{\delta }_2,{\delta }_3)=(1.5,1.4,2.1,2.5,-0.4,-0.7,-0.2)$. The JPDF (14-a) is defined only in a small closed interval $(0.6,0.6)⪅(x,y)⪅(1,1)$ is plotted with JCDF (14-b) as it approaches to 1 for high values of $(x,y)$. Statistical quantities depicted in Figure 15, Figure 16 suggests applying this model for ill-conditioned, positive small, and right skewed data.

Figure 14.

Bivariate JPDF and JCDF of $BIq-\mathcal{G}\mathcal{E}\mathcal{V}$ distribution for Case-3.

Figure 15.

Bivariate hazard function with correlation structures for (u,v)∈(1,1) of $BIq-\mathcal{G}\mathcal{E}\mathcal{V}$ distribution for Case-3.

Figure 16.

Marginal PDFs and CDFs of $BIq-\mathcal{G}\mathcal{E}\mathcal{V}$ distribution for Case-3.

Case 4: For the parameters $(s,m,\xi ,q,{\delta }_1,{\delta }_2,{\delta }_3)=(0.5,0.8,0.2,0.4,0.4,0.7,0.2)$, consider the Figure 17, Figure 18, Figure 19. We obtain another shapes for the JPDF in (17-a) defined for $(x,y)⪆(-5,5)$ with left tail in agreement with JCDF (17-b). Other corresponding statistical measures are presented in (17-c)-(17-e). The marginal densities (17-f)-(17-i) seem fit to model right skewed data.

Figure 17.

Bivariate JPDF and JCDF of $BIq-\mathcal{G}\mathcal{E}\mathcal{V}$ distribution for Case-4.

Figure 18.

Case-4 bivariate statistical measures for (−5,5) domain of $BIq-\mathcal{G}\mathcal{E}\mathcal{V}$ distribution.

Figure 19.

Marginal density of $BIq-\mathcal{G}\mathcal{E}\mathcal{V}$ distribution for Case-4.

Case 5: The bi-variate $q-Gumbel$ is considered in Figure 20, Figure 21, Figure 22 for the parameters $(s,m,q,{\delta }_1,{\delta }_2,{\delta }_3)=$ $(0.6,1.2,0.8,0.6,0.4,0.3)$. The symmetry of the JPDF (20-a) and marginal PDFs (20-f);(20-h) is graphically evident. The JCDF (21-b) and marginal CDFs are given (21-g);(21-i). The Fig. 22 (22:c-e) signify the plots of bivariate hazard function along with the respective correlation structure.

Figure 20.

Bivariate JPDF with marginal densities of q − Gumbel in Case-5.

Figure 21.

Bivariate JCDF with marginal CDFs of q − Gumbel in Case-5.

Figure 22.

Some more bivariate statistical measures of q − Gumbel in Case-5.

3.4. Climate data

One of the major impacts of climate change is the global wind pattern cause alterations in average temperatures of regions. For example, the changes in mean temperatures in Asia can disturb the balance between land and ocean, causing a major shift in intensity or duration of monsoon weather system. This altered rainfall paradigm will have significant effect on the crop yields leading to ultimate concerns on food security. This will spearhead a cascade effect on the environmental and human aspect of societies. The model understudy can help us understand the impact of climate change more productively. Consider the following climate data consisting of the normal temperature ${}^{\circ }C$ and the mean wind speed $km/h$ from 1-1-1995 to 28-2-1995 for the Madrid/Barajas station to be treated as a bi-variate random variable $(X,Y)\sim BIq-GEV(s,m,\xi ,q,{\delta }_1,{\delta }_2,{\delta }_3)$. Typically, the greater the temperature differential, the stronger the winds that develop. A local airflow that affects winds can also be driven on by variations in temperature between the land and the ocean. The data is accessible at https://en.tutiempo.net/climate. Table 3 presents the estimated values for the unknown parameters and the related information criterions which includes AIC and BIC. Some related visualizations are also presented in Figure 23, Figure 24, Figure 25.

Table 3.

The estimates of (s,m,ξ,q,δ₁,δ₂,δ₃).

Parameter	Estimate	AIC	BIC
S	0.21514177
m	0.74469873
ξ	0.05098449
q	-0.47074546	730.6066	745.1493621
σ1	-0.13546057
σ2	0.14012899
σ2	-0.53850799

Figure 23.

Bivariate JPDF, JCDF and hazard function based on the estimates in Table 3.

Figure 24.

Some other measures based on the estimates in Table 3.

Figure 25.

The fitted marginal PDFs and CDFs of BIq − Gumbel model.

4. Conclusion

This article sets out to argue the importance of bivariate modelling related to events triggered due to climate change data. A bivariate extension of $q-\mathcal{G}\mathcal{E}\mathcal{V}$, with q- Gumbel distribution as special case, is conceptualized using a novel as well as traditional approach. A technique is developed from independent $q-\mathcal{G}\mathcal{E}\mathcal{V}$ random variables to model an associated bivariate structures arising of extreme values in terms of $q-\mathcal{G}\mathcal{E}\mathcal{V}$ probability density functions. Further, a bivariate flood data set has been employed to model bivariate $q-\mathcal{G}\mathcal{E}\mathcal{V}$ via established goodness of fit measures. Additionally, for bivariate probability models to properly estimate the parameters, there must be enough data. Insufficient data or strongly correlated variables can culminate incorrect estimates and suboptimal model performance. In comparison to competing models, the empirical findings hence yielded, provide a much superior fit to the data under consideration. Apart from that, a traditional method for incorporating bivariate extension to the univariate $q-\mathcal{G}\mathcal{E}\mathcal{V}$ distribution has also been addressed. The related inference has been carried out using the method of maximum likelihood estimation supported by its applicability on climate data. Given the significance of unique methodology being introduced in this article, we hope that this can be extended to establish multivariate $q-\mathcal{G}\mathcal{E}\mathcal{V}$ distribution. Additionally, this bivariate extension can also be utilized to study the joint occurrences of extreme random variables in vast scientific domains. The rank-based inference of the presented bivariate $q-\mathcal{G}\mathcal{E}\mathcal{V}$ model parameters could well be explored for future considerations.

CRediT authorship contribution statement

Laila A. Al-Essa: Funding acquisition, Conceptualization. Abdus Saboor: Writing – original draft, Methodology, Investigation, Formal analysis, Conceptualization. Muhammad H. Tahir: Supervision, Resources. Sadaf Khan: Writing – review & editing, Visualization, Validation, Investigation, Data curation. Farrukh Jamal: Validation, Software, Investigation. Ahmed Elhassanein: Writing – original draft, Visualization, Methodology, Formal analysis.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

The selection of data cited to corroborate this article's conclusion can be accessed at https://en.tutiempo.net/climate.