# A Generator of Bivariate Distributions: Properties, Estimation, and Applications

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. The GBD Family

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Theorem**

**3.**

## 3. Special Cases

**Extended bivariate generalized exponential model.**A random variable U follows a GE distribution, $U\sim GE(\theta ,\lambda )$ (see Gupta and Kundu [25]), if its cdf is given by

**Extended bivariate proportional reversed hazard rate model.**If ${U}_{i}\sim PRH\left({\theta}_{i}\right)$ with base distribution ${F}_{{B}_{i}}$ $i=1,2,3$, i.e., its cdf can be expressed as ${F}_{{U}_{i}}={F}_{{B}_{i}}^{{\theta}_{i}}$ (see Gupta et al. [22] and Di Crescenzo [23]), then the GBD model with PRH baseline distribution vector provides an extended BPRH model, $({X}_{1},{X}_{2})\sim EBPRH(\mathit{\theta},\mathit{\lambda})$, with $\mathit{\theta}=({\theta}_{1},{\theta}_{2},{\theta}_{3})$ parameter vector of the PRH components and $\mathit{\lambda}=({\lambda}_{1},{\lambda}_{2},{\lambda}_{3})$ parameter vector of the underlying distributions ${F}_{{B}_{i}}$’s. From (1), its joint cdf is given by

**Extended bivariate generalized linear failure rate model.**It is said that a random variable U follows a GLFR distribution, $U\sim GLFR(\theta ,\lambda ,\gamma )$ (see Sarhan and Kundu [26]), if its cdf is given by

**Extended bivariate log-exponentiated Kumaraswamy model.**Let U be a random variable with logEK distribution, $U\sim logEK(\theta ,\lambda ,\gamma )$ (see Lemonte et al. [27]), then its cdf

**Extended bivariate exponentiated modified Weibull extension model.**A random variable U follows an EMWE distribution, $U\sim EMWE(\theta ,\alpha ,\beta ,\lambda )$ (see Sarhan and Apaloo [28]), if its cdf can be expressed as

**Extended bivariate inverse Weibull model.**The cdf of the IW distribution (e.g., see Keller et al. [29]) is defined by

**Extended bivariate Dagum model.**It is said that a random variable U follows a Dagum distribution [30], $U\sim D(\theta ,\lambda ,\gamma )$, if its cdf is given by

**Extended bivariate generalized Rayleigh model.**The cdf of the GR distribution, also called Burr type X model [31], is

**Extended bivariate Gumbel-G model.**Alzaatrech et al. [32] proposed a transformed-transformer method for generating families of continuous distributions. From such method, it is said that a random variable U follows a Gumbel-G model, $U\sim Gu$-$G(\theta ,\alpha ,\mathit{\lambda})$ if its cdf can be expressed as

**Extended bivariate generalized inverted Kumaraswamy model.**A random variable U is said to be a GIK distribution defined by Iqbal et al. [33], if its cdf is given by

**Extended bivariate Burr type X-G model.**From the transformed-transformer method of Alzaatrech et al. [32], it is said that a random variable U follows a Burr X-G model, $U\sim BX$-$G(\theta ,\mathit{\lambda})$ if its cdf can be expressed as

**GBD models from different baseline components.**In addition, a GBD model can be derived from baseline components ${U}_{i}$s belonging to different distribution families, which allows one to generate new bivariate distributions.

## 4. Distributional Properties

#### 4.1. Marginal and Conditional Distributions

**Proposition**

**1.**

**Proof.**

**Corollary**

**1.**

**Theorem**

**4.**

- 1.
- The conditional distribution of ${X}_{i}$ given ${X}_{j}\le {x}_{j}$ ($i\ne j$), say ${F}_{i|{X}_{j}\le {x}_{j}}$, is an absolutely continuous cdf given by$${F}_{i|{X}_{j}\le {x}_{j}}\left({x}_{i}\right)=\left(\right)open="\{"\; close>\begin{array}{cc}{F}_{{U}_{i}}\left({x}_{i}\right)\frac{{F}_{{U}_{3}}\left({x}_{i}\right)}{{F}_{{U}_{3}}\left({x}_{j}\right)},\hfill & if\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4pt}{0ex}}{x}_{i}{x}_{j}\hfill \\ {F}_{{U}_{i}}\left({x}_{i}\right),\hfill & if\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4pt}{0ex}}{x}_{i}\ge {x}_{j}\hfill \end{array}$$
- 2.
- The conditional pdf of ${X}_{i}$ given ${X}_{j}={x}_{j}$ ($i\ne j$), say ${f}_{i|{X}_{j}={x}_{j}}$, is a convex combination of an absolutely continuous cdf and a degenerate cdf given by$${f}_{i|{X}_{j}={x}_{j}}\left({x}_{i}\right)={\alpha}_{j}{I}_{{x}_{j}}\left({x}_{i}\right)+(1-{\alpha}_{j}){f}_{i|{x}_{j},ac}\left({x}_{i}\right),$$$${f}_{i|{X}_{j}={x}_{j},ac}\left({x}_{i}\right)=\frac{1}{1-{\alpha}_{j}}\left(\right)open="\{"\; close>\begin{array}{cc}{f}_{{X}_{i}}\left({x}_{i}\right)\frac{{f}_{{U}_{j}}\left({x}_{j}\right)}{{f}_{{X}_{j}}\left({x}_{j}\right)},\hfill & if\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4pt}{0ex}}{x}_{i}{x}_{j}\hfill \\ {f}_{{U}_{i}}\left({x}_{i}\right),\hfill & if\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4pt}{0ex}}{x}_{i}{x}_{j}\hfill \\ 0,\hfill & if\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4pt}{0ex}}{x}_{i}={x}_{j}\hfill \end{array}$$$${\alpha}_{j}={F}_{{U}_{1}}\left({x}_{j}\right){F}_{{U}_{2}}\left({x}_{j}\right)\frac{{f}_{{U}_{3}}\left({x}_{j}\right)}{{f}_{{X}_{j}}\left({x}_{j}\right)}.$$

#### 4.2. Minimum and Maximum Order Statistics

**Theorem**

**5.**

**Proof.**

**Corollary**

**2.**

## 5. Dependence and Stochastic Properties

#### 5.1. GBD Model

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

- 1.
- $({X}_{1},{X}_{2})$ is left tail decreasing (LTD).
- 2.
- $({X}_{1},{X}_{2})$ is left corner set decreasing (LCSD).
- 3.
- Its joint cdf F is totally positive and of order 2 ($T{P}_{2}$).

**Proof.**

**Proposition**

**4.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**Corollary**

**3.**

**Proof.**

**Remark**

**1.**

#### 5.2. Marginals and Order Statistics

**Theorem**

**8.**

**Remark**

**2.**

**Example**

**1.**

**Example**

**2.**

**Remark**

**3.**

**Remark**

**4.**

**Example**

**3.**

**Example**

**4.**

**Theorem**

**9.**

- 1.
- If ${U}_{i}{\le}_{st}{V}_{i}$ ($i=1,2,3$), then ${X}_{i}{\le}_{st}{Y}_{i}$ ($i=1,2$).
- 2.
- If ${U}_{i}{\le}_{rh}{V}_{i}$ ($i=1,2,3$), then ${X}_{i}{\le}_{rh}{Y}_{i}$ ($i=1,2$).

**Theorem**

**10.**

**Example**

**5.**

**Example**

**6.**

**Remark**

**5.**

**Remark**

**6.**

**Example**

**7.**

**Theorem**

**11.**

**Corollary**

**4.**

**Example**

**8.**

**Remark**

**7.**

**Theorem**

**12.**

#### 5.3. Copula and Related Association Measures

**Theorem**

**13.**

**Proof.**

**Corollary**

**5.**

**Corollary**

**6.**

**Corollary**

**7.**

**Kendall’s tau.**The Kendall’s $\tau $ is defined as the probability of concordance minus the probability of discordance between two pairs of independent and identically distributed random vectors, $({X}_{1},{X}_{2})$ and $({Y}_{1},{Y}_{2})$, as follows:

**Spearman’s rho.**The Spearman’s $\rho $ coefficient measures the dependence by three pairs of independent and identically distributed random vectors, $({X}_{1},{X}_{2})$, $({Y}_{1},{Y}_{2})$ and $({Z}_{1},{Z}_{2})$. It is defined as

**Blomqvist’s Beta.**The Blomqvist’s $\beta $ coefficient, also called the medial correlation coefficient, is defined as the probability of concordance minus the probability of discordance between $({X}_{1},{X}_{2})$ and its median point, say $({m}_{1},{m}_{2})$, taking the following form:

**Tail Dependence.**The tail dependence measures the association of extreme events in both directions, the upper (lower) tail dependence ${\lambda}_{U}$ (${\lambda}_{L}$) provides an asymptotical association measurement in the upper (lower) quadrant tail of a bivariate random vector, given by (if it exists)

## 6. Maximum Likelihood Estimation

- If $i\in {I}_{0}$, then$${u}_{jim}\left(\mathit{\theta}\right)=E\left({U}_{j}\right|{U}_{j}<{x}_{i})=\frac{1}{{F}_{{U}_{j}}\left({x}_{i}\right)}{\int}_{-\infty}^{{x}_{i}}u{f}_{{U}_{j}}\left(u\right)du,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}j=1,2.$$
- If $i\in {I}_{1}$ and $j,k\in \{1,3\},j\ne k$, then$$\begin{array}{cc}\hfill \phantom{\rule{-15.0pt}{0ex}}{u}_{jim}\left(\mathit{\theta}\right)& =E\left({U}_{j}\right|max\{{U}_{1},{U}_{3}\}={x}_{1i})\hfill \\ \hfill & ={x}_{1i}P({U}_{j}>{U}_{k})+P({U}_{j}<{U}_{k})\frac{1}{{F}_{{U}_{j}}\left({x}_{1i}\right)}{\int}_{-\infty}^{{x}_{1i}}u{f}_{{U}_{j}}\left(u\right)du\hfill \\ \hfill & ={x}_{1i}{\int}_{-\infty}^{\infty}{f}_{{U}_{j}}\left(u\right){F}_{{U}_{k}}\left(u\right)du+\frac{1}{{F}_{{U}_{j}}\left({x}_{1i}\right)}{\int}_{-\infty}^{\infty}{f}_{{U}_{k}}\left(u\right){F}_{{U}_{j}}\left(u\right)du\phantom{\rule{4pt}{0ex}}{\int}_{-\infty}^{{x}_{1i}}u{f}_{{U}_{j}}\left(u\right)du.\hfill \end{array}$$
- If $i\in {I}_{2}$ and $j,k\in \{2,3\},j\ne k$, then$$\begin{array}{cc}\hfill \phantom{\rule{-15.0pt}{0ex}}{u}_{jim}\left(\mathit{\theta}\right)& =E\left({U}_{j}\right|max\{{U}_{2},{U}_{3}\}={x}_{2i})\hfill \\ \hfill & ={x}_{2i}P({U}_{j}>{U}_{k})+P({U}_{j}<{U}_{k})\frac{1}{{F}_{{U}_{j}}\left({x}_{2i}\right)}{\int}_{-\infty}^{{x}_{2i}}u{f}_{{U}_{j}}\left(u\right)du\hfill \\ \hfill & ={x}_{2i}{\int}_{-\infty}^{\infty}{f}_{{U}_{j}}\left(u\right){F}_{{U}_{k}}\left(u\right)du+\frac{1}{{F}_{{U}_{j}}\left({x}_{2i}\right)}{\int}_{-\infty}^{\infty}{f}_{{U}_{k}}\left(u\right){F}_{{U}_{j}}\left(u\right)du\phantom{\rule{4pt}{0ex}}{\int}_{-\infty}^{{x}_{2i}}u{f}_{{U}_{j}}\left(u\right)du.\hfill \end{array}$$

**E-step**

- At the k-th step for $i\in {I}_{0}$, obtain the missing ${u}_{1i}$ and ${u}_{2i}$ as ${u}_{1im}\left({\mathit{\theta}}^{\left(k\right)}\right)$ and ${u}_{2im}\left({\mathit{\theta}}^{\left(k\right)}\right)$, respectively. For $i\in {I}_{1}$ obtain the missing ${u}_{1i}$ and ${u}_{3i}$ as ${u}_{1im}\left({\mathit{\theta}}^{\left(k\right)}\right)$ and ${u}_{3im}\left({\mathit{\theta}}^{\left(k\right)}\right)$, respectively. Similarly, for $i\in {I}_{2}$, obtain the missing ${u}_{2i}$ and ${u}_{3i}$ as ${u}_{2im}\left({\mathit{\theta}}^{\left(k\right)}\right)$ and ${u}_{3im}\left({\mathit{\theta}}^{\left(k\right)}\right)$, respectively.
- Form the ’pseudo’ log-likelihood function as ${\ell}_{s}^{\left(k\right)}\left(\mathit{\theta}\right)={\ell}_{1s}^{\left(k\right)}\left({\mathit{\theta}}_{1}\right)+{\ell}_{2s}^{\left(k\right)}\left({\mathit{\theta}}_{2}\right)+{\ell}_{3s}^{\left(k\right)}\left({\mathit{\theta}}_{3}\right)$, where$$\begin{array}{cc}\hfill \phantom{\rule{-10.0pt}{0ex}}{\ell}_{1s}^{\left(k\right)}\left({\mathit{\theta}}_{1}\right)& =\sum _{i\in {I}_{0}}ln{f}_{{U}_{1}}({u}_{1im}\left({\mathit{\theta}}^{\left(k\right)}\right);{\mathit{\theta}}_{1})+\sum _{i\in {I}_{1}}ln{f}_{{U}_{1}}({u}_{1im}\left({\mathit{\theta}}^{\left(k\right)}\right);{\mathit{\theta}}_{1})+\sum _{i\in {I}_{2}}ln{f}_{{U}_{1}}({u}_{1i};{\mathit{\theta}}_{1})\hfill \\ \hfill \phantom{\rule{-10.0pt}{0ex}}{\ell}_{2s}^{\left(k\right)}\left({\mathit{\theta}}_{2}\right)& =\sum _{i\in {I}_{0}}ln{f}_{{U}_{2}}({u}_{2im}\left({\mathit{\theta}}^{\left(k\right)}\right);{\mathit{\theta}}_{2})+\sum _{i\in {I}_{1}}ln{f}_{{U}_{2}}({u}_{2i};{\mathit{\theta}}_{2})+\sum _{i\in {I}_{2}}ln{f}_{{U}_{2}}({u}_{2im}\left({\mathit{\theta}}^{\left(k\right)}\right);{\mathit{\theta}}_{2})\hfill \\ \hfill \phantom{\rule{-10.0pt}{0ex}}{\ell}_{3s}^{\left(k\right)}\left({\mathit{\theta}}_{3}\right)& =\sum _{i\in {I}_{0}}ln{f}_{{U}_{3}}({u}_{3i};{\mathit{\theta}}_{3})+\sum _{i\in {I}_{1}}ln{f}_{{U}_{3}}({u}_{3im}\left({\mathit{\theta}}^{\left(k\right)}\right);{\mathit{\theta}}_{3})+\sum _{i\in {I}_{2}}ln{f}_{{U}_{3}}({u}_{3im}\left({\mathit{\theta}}^{\left(k\right)}\right);{\mathit{\theta}}_{3}).\hfill \end{array}$$

**M-step**

- ${\mathit{\theta}}^{(k+1)}=({\mathit{\theta}}_{1}^{(k+1)},{\mathit{\theta}}_{2}^{(k+1)},{\mathit{\theta}}_{3}^{(k+1)})$ can be obtained by maximizing ${\ell}_{1s}^{\left(k\right)}\left({\mathit{\theta}}_{1}\right)$, ${\ell}_{2s}^{\left(k\right)}\left({\mathit{\theta}}_{2}\right)$ and ${\ell}_{3s}^{\left(k\right)}\left({\mathit{\theta}}_{3}\right)$ with respect to ${\mathit{\theta}}_{1}$, ${\mathit{\theta}}_{2}$ and ${\mathit{\theta}}_{3}$, respectively.

## 7. Data Analysis

#### 7.1. Soccer Data

**Example**

**9.**

**Example**

**10.**

#### 7.2. Diabetic Retinopathy Data

**Example**

**11.**

**Example**

**12.**

## 8. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Proof of**

**Theorem**

**2.**

**Proof of**

**Theorem**

**3.**

**Proof of**

**Theorem**

**4.**

**Proof of**

**Theorem**

**11.**

**Proof of**

**Corollary**

**4.**

## Appendix B

**Model I.**

- If $i\in {I}_{0}$, then$$\begin{array}{cc}\hfill {u}_{1im}\left(\mathit{\theta}\right)& =E\left({U}_{1}\right|{U}_{1}<{x}_{i})=H({x}_{i};{\lambda}_{1})\hfill \\ \hfill {u}_{2im}\left(\mathit{\theta}\right)& =E\left({U}_{2}\right|{U}_{2}<{x}_{i})=H({x}_{i};{\lambda}_{2}),\hfill \end{array}$$$$H(x;\lambda )=\frac{1}{\lambda}-\frac{x{e}^{-\lambda x}}{1-{e}^{-\lambda x}}.$$
- If $i\in {I}_{1}$, then$$\begin{array}{cc}\hfill {u}_{1im}\left(\mathit{\theta}\right)& =E\left({U}_{1}\right|max\{{U}_{1},{U}_{3}\}={x}_{1i})=\frac{{\lambda}_{3}}{{\lambda}_{1}+{\lambda}_{3}}{x}_{1i}+\frac{{\lambda}_{1}}{{\lambda}_{1}+{\lambda}_{3}}H({x}_{1i};{\lambda}_{1})\hfill \\ \hfill {u}_{3im}\left(\mathit{\theta}\right)& =E\left({U}_{3}\right|max\{{U}_{1},{U}_{3}\}={x}_{1i})=\frac{{\lambda}_{1}}{{\lambda}_{1}+{\lambda}_{3}}{x}_{1i}+\frac{{\lambda}_{3}}{{\lambda}_{1}+{\lambda}_{3}}H({x}_{1i};{\lambda}_{3}).\hfill \end{array}$$
- If $i\in {I}_{2}$, then$$\begin{array}{cc}\hfill {u}_{2im}\left(\mathit{\theta}\right)& =E\left({U}_{2}\right|max\{{U}_{2},{U}_{3}\}={x}_{2i})=\frac{{\lambda}_{3}}{{\lambda}_{2}+{\lambda}_{3}}{x}_{2i}+\frac{{\lambda}_{2}}{{\lambda}_{2}+{\lambda}_{3}}H({x}_{2i};{\lambda}_{2})\hfill \\ \hfill {u}_{3im}\left(\mathit{\theta}\right)& =E\left({U}_{3}\right|max\{{U}_{2},{U}_{3}\}={x}_{2i})=\frac{{\lambda}_{2}}{{\lambda}_{2}+{\lambda}_{3}}{x}_{2i}+\frac{{\lambda}_{3}}{{\lambda}_{2}+{\lambda}_{3}}H({x}_{2i};{\lambda}_{3}).\hfill \end{array}$$

**Model II.**

- If $i\in {I}_{0}$, then$${u}_{jim}\left(\mathit{\theta}\right)=E\left({U}_{j}\right|{U}_{j}<{x}_{i})={H}_{W}({x}_{i};{\alpha}_{j},{\lambda}_{j}),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}j=1,2,$$$${H}_{W}(x;\alpha ,\lambda )=\frac{1}{1-{e}^{-\lambda {x}^{\alpha}}}{\int}_{0}^{\lambda {x}^{\alpha}}{\left(\right)}^{\frac{u}{\lambda}}1/\alpha $$
- If $i\in {I}_{1}$, then$$\begin{array}{cc}\hfill {u}_{1im}\left(\mathit{\theta}\right)& =E\left({U}_{1}\right|max\{{U}_{1},{U}_{3}\}={x}_{1i})={p}_{13}{x}_{1i}+(1-{p}_{13}){H}_{W}({x}_{1i};{\alpha}_{1},{\lambda}_{1})\hfill \\ \hfill {u}_{3im}\left(\mathit{\theta}\right)& =E\left({U}_{3}\right|max\{{U}_{1},{U}_{3}\}={x}_{1i})=(1-{p}_{13}){x}_{1i}+{p}_{13}{H}_{G}({x}_{1i};{\alpha}_{3},{\lambda}_{3}),\hfill \end{array}$$$$K(\alpha ,\lambda )={\int}_{0}^{\infty}\alpha \lambda {x}^{\alpha -1}{e}^{-\lambda {x}^{\alpha}}{(1-{e}^{-{\lambda}_{3}x})}^{{\alpha}_{3}}dx,\phantom{\rule{4pt}{0ex}}{H}_{G}(x;\alpha ,\lambda )=x-\frac{1}{\lambda {(1-{e}^{-\lambda x})}^{\alpha}}{\int}_{0}^{1-{e}^{-\lambda x}}\frac{{t}^{\alpha}}{1-t}dt$$
- If $i\in {I}_{2}$, then$$\begin{array}{cc}\hfill {u}_{2im}\left(\mathit{\theta}\right)& =E\left({U}_{2}\right|max\{{U}_{2},{U}_{3}\}={x}_{2i})={p}_{23}{x}_{2i}+(1-{p}_{23}){H}_{W}({x}_{2i};{\alpha}_{2},{\lambda}_{2})\hfill \\ \hfill {u}_{3im}\left(\mathit{\theta}\right)& =E\left({U}_{3}\right|max\{{U}_{2},{U}_{3}\}={x}_{2i})=(1-{p}_{23}){x}_{2i}+{p}_{23}{H}_{G}({x}_{2i};{\alpha}_{3},{\lambda}_{3}),\hfill \end{array}$$

## Appendix C

**Table A1.**Summary of fitted GBD models for the two real data. EM rows are the parameters estimated with the EM algorithm for maximizing the pseudo log-likelihood function, along with the log-likelihood, AIC and BIC values, and BFGS rows correspond to the results obtained by applying the Broyden–Fletcher–Goldfarb–Shanno algorithm for maximizing the log-likelihood function.

GBD Model | $\mathit{\theta}$ | $\mathit{\ell}\left(\mathit{\theta}\right)$ | AIC | BIC | |||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{\alpha}}_{1}$ | ${\mathit{\lambda}}_{1}$ | ${\mathit{\alpha}}_{2}$ | ${\mathit{\lambda}}_{2}$ | ${\mathit{\alpha}}_{3}$ | ${\mathit{\lambda}}_{3}$ | ||||

Soccer data | |||||||||

Model I | |||||||||

EM | 0.03126 | 0.04630 | 0.04269 | −299.4331 | 604.8663 | 609.6990 | |||

BFGS | 0.03116 | 0.04636 | 0.04283 | −299.4328 | 604.8656 | 609.6984 | |||

Model II | |||||||||

EM | 1.2987 | 0.0097 | 0.8047 | 0.0093 | 1.0037 | 0.0369 | −348.2715 | 708.5430 | 718.2085 |

BFGS | 1.3808 | 0.00698 | 0.5652 | 0.25469 | 1.53813 | 0.05219 | −295.3057 | 602.6114 | 612.2770 |

Diabetic retinopathy data | |||||||||

Model I | |||||||||

EM | 0.0653 | 0.0737 | 0.1345 | −289.9878 | 585.9757 | 590.8884 | |||

BFGS | 0.06290 | 0.07181 | 0.14282 | −289.9144 | 585.8288 | 590.7415 | |||

Model II | |||||||||

EM | 1.0937 | 0.0447 | 0.5851 | 0.2369 | 0.8995 | 0.1898 | −290.0758 | 592.1515 | 601.9770 |

BFGS | 1.1477 | 0.03920 | 0.7917 | 0.13923 | 0.41913 | 0.08272 | −285.5795 | 583.1590 | 592.9846 |

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**Figure 1.**Surface and contour plots of the joint pdf of GBD models $({X}_{1},{X}_{2})$ with different components $({U}_{1},{U}_{2},{U}_{3})$.

**Figure 2.**Plots of the marginal pdfs of the GBD models $({X}_{1},{X}_{2})$ with different components $({U}_{1},{U}_{2},{U}_{3})$.

${\mathit{I}}_{\mathit{k}}$ | Ordering of ${\mathit{U}}_{\mathit{j}}$ | ${\mathit{X}}_{1}$ | ${\mathit{X}}_{2}$ | Missing |
---|---|---|---|---|

${I}_{0}$ | ${u}_{1i}<{u}_{2i}<{u}_{3i}$ | ${u}_{3i}$ | ${u}_{3i}$ | ${u}_{1i}$, ${u}_{2i}$ |

${I}_{0}$ | ${u}_{2i}<{u}_{1i}<{u}_{3i}$ | ${u}_{3i}$ | ${u}_{3i}$ | ${u}_{1i}$, ${u}_{2i}$ |

${I}_{1}$ | ${u}_{1i}<{u}_{3i}<{u}_{2i}$ | ${u}_{3i}$ | ${u}_{2i}$ | ${u}_{1i}$ |

${I}_{1}$ | ${u}_{3i}<{u}_{1i}<{u}_{2i}$ | ${u}_{1i}$ | ${u}_{2i}$ | ${u}_{3i}$ |

${I}_{2}$ | ${u}_{2i}<{u}_{3i}<{u}_{1i}$ | ${u}_{1i}$ | ${u}_{3i}$ | ${u}_{2i}$ |

${I}_{2}$ | ${u}_{3i}<{u}_{2i}<{u}_{1i}$ | ${u}_{2i}$ | ${u}_{1i}$ | ${u}_{3i}$ |

GBD Model | KS (p-Value) | ||||
---|---|---|---|---|---|

${\mathit{X}}_{1}$ | ${\mathit{X}}_{2}$ | $max\{{\mathit{X}}_{1},{\mathit{X}}_{2}\}$ | AIC | BIC | |

Model I | 0.1491 (0.3830) | 0.1099 (0.7622) | 0.1530 (0.3517) | 604.8663 | 609.6990 |

Model II | 0.0976 (0.8719) | 0.0839 (0.9565) | 0.1139 (0.7228) | 708.5430 | 718.2085 |

GBD Model | KS (p-Value) | ||||
---|---|---|---|---|---|

${\mathit{X}}_{1}$ | ${\mathit{X}}_{2}$ | $max\{{\mathit{X}}_{1},{\mathit{X}}_{2}\}$ | AIC | BIC | |

Model I | 0.1033 (0.8244) | 0.1848 (0.1598) | 0.1229 (0.6310) | 585.9757 | 590.8884 |

Model II | 0.0920 (0.8960) | 0.0952 (0.8706) | 0.1152 (0.6778) | 592.1515 | 601.9770 |

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**MDPI and ACS Style**

Franco, M.; Vivo, J.-M.; Kundu, D.
A Generator of Bivariate Distributions: Properties, Estimation, and Applications. *Mathematics* **2020**, *8*, 1776.
https://doi.org/10.3390/math8101776

**AMA Style**

Franco M, Vivo J-M, Kundu D.
A Generator of Bivariate Distributions: Properties, Estimation, and Applications. *Mathematics*. 2020; 8(10):1776.
https://doi.org/10.3390/math8101776

**Chicago/Turabian Style**

Franco, Manuel, Juana-María Vivo, and Debasis Kundu.
2020. "A Generator of Bivariate Distributions: Properties, Estimation, and Applications" *Mathematics* 8, no. 10: 1776.
https://doi.org/10.3390/math8101776