# A New Class of Alternative Bivariate Kumaraswamy-Type Models: Properties and Applications

## Abstract

**:**

## 1. Introduction

- Model $1:$ In this case, we assume the joint density will be of the form$$f\left(x,y\right)=D\alpha \beta \frac{{x}^{\alpha -1}{y}^{\beta -1}{\left(1-{x}^{\alpha}\right)}^{\theta +\delta -1}{\left(1-{y}^{\beta}\right)}^{\theta +\delta -1}}{{\left[1-\theta {x}^{\alpha}{y}^{\beta}\right]}^{\delta}},I\left(0<x<1,0<y<1\right),$$$$D={\left[{\int}_{0}^{1}{\int}_{0}^{1}\alpha \beta \frac{{x}^{\alpha -1}{y}^{\beta -1}{\left(1-{x}^{\alpha}\right)}^{\theta +\delta -1}{\left(1-{y}^{\beta}\right)}^{\theta +\delta -1}}{{\left[1-\theta {x}^{\alpha}{y}^{\beta}\right]}^{\delta}}dxdy\right]}^{-1}.$$In this case, the parametric restrictions are the following: $\theta +\delta >1$, $0\le \theta <1$, $\alpha >0$ and $\beta >0$. The parameters $\delta $ and $\theta $ influence the correlation measure between the X and Y components of the distribution. Note that when $\theta =0$, the joint density reduces to the product of two independent univariate KW random variables, particularly when $\theta =0,$$X\sim KW\left(\alpha ,\theta +\delta \right)$ and $Y\sim KW\left(\beta ,\theta +\delta \right)$ independently. Since the KW distribution has two shape parameters, it appears that the marginals of both X and Y have the same second shape parameter, specifically $\left(\theta +\delta \right)$, but different first shape parameters, which are $\alpha $ and $\beta $, respectively. Potential application of such bivariate probability models can be envisioned in real-life scenarios where, for example, X and Y have data structures such that one characteristic is common to both of them, but the other one is different. The only factor that might work as a deterrent regarding the flexibility of such a model is how a practitioner can guarantee that restrictions such as $\theta +\delta >1$ and $0\le \theta <1$ would be met in reality. Notice that since $\theta <1$ and $\delta >1-\theta ,$ one can easily observe that $0<\delta <1.$ Additional discussion of the parameters is given in the structural properties section later. However, one may consider appropriate testing of the hypothesis as a part of model fitting regarding whether these parametric constraints are met.In the next section, we consider another bivariate KW-type model that we conjecture to have some flexibility in terms of modeling positively dependent data.
- Model $2:$ Suppose that the joint density is of the form$$\begin{array}{ccc}\hfill f\left(x,y\right)& =& C\alpha \beta {\theta}_{1}{\theta}_{2}{\gamma}_{1}{\gamma}_{2}{x}^{\alpha -1}{y}^{\beta -1}{\left[1-{\theta}_{1}{x}^{\alpha}-{\theta}_{2}{y}^{\beta}\right]}^{\delta -\alpha -\beta -1}{\left(1-{\theta}_{1}{x}^{\alpha}\right)}^{{\gamma}_{1}-1}{\left(1-{\theta}_{2}{y}^{\beta}\right)}^{{\gamma}_{2}-1}\hfill \\ & & \times I\left(0<x<1,0<y<1\right),\hfill \end{array}$$Noticeably, when $\delta =\alpha +\beta +1,$ it appears that ${\theta}_{1}X\sim KW\left(\alpha ,{\gamma}_{1}\right),$ and ${\theta}_{2}Y\sim KW\left(\beta ,{\gamma}_{2}\right)$ independently. Therefore, by invoking independence via a linear constraint on the parameters, component-wise, the marginals of both X and Y follow a scaled version of a two-parameter univariate KW distribution with the scale factors ${\theta}_{1}$ and ${\theta}_{2}$, respectively. Furthermore, this model is significantly different from the first proposed model in the sense that we bring in scale factors to capture the variability of X and $Y,$ and the shape parameters are different. This feature is different from the first model, in which the second shape parameter is the same for both X and $Y.$ Once again, as before, a natural objection that might occur in terms of application in modeling real-life data based on this probability model is how our informed expert can guarantee that for dependence modeling, the linear restriction is not satisfied. One simple strategy would be to simply consider a hypothesis test of independence, which can be written as follows:${H}_{0}:\delta =\alpha +\beta +1,$ against the alternative ${H}_{a}:\delta \ne \alpha +\beta +1.$Some additional discussion on the parameters is given in the structural properties section for this model later.

- (1)
- The Gauss hypergeometric function is defined by$${}_{2}F_{1}\left(a,b;c;x\right)=\sum _{k=0}^{\infty}\frac{{\left(a\right)}_{k}{\left(b\right)}_{k}}{{\left(c\right)}_{k}}\frac{{x}^{k}}{k!},$$
- (2)
- The series expansion is given by$${(1-z)}^{-\delta}=\sum _{j=0}^{\infty}\frac{\Gamma (\delta +j){z}^{j}}{\Gamma \left(\delta \right)j!}.$$

## 2. Bivariate KW-Type Model 1 Structural Properties

#### 2.1. Marginal Densities

`Mathematica`and the Gauss hypergeometric function given in Equation (3).

`Mathematica`and Equation (3) again.

- Observe that as $x\to 0,$$f\left(x,y\right)\sim D\alpha \beta {x}^{\alpha -1}{y}^{\beta -1}{\left(1-{y}^{\beta}\right)}^{\theta +\delta -1}.$ Thus, the distribution of Y along the vertical boundary of the simplex belongs to a univariate KW distribution with two shape parameters: $\beta $ and $\theta +\delta .$ The parameter $\alpha $ represents the scale variation from a standard KW model.
- Likewise, as $y\to 0,$$f\left(x,y\right)\sim D\alpha \beta {x}^{\alpha -1}{y}^{\beta -1}{\left(1-{x}^{\alpha}\right)}^{\theta +\delta -1}.$ Therefore, the distribution of X along the vertical boundary of the simplex belongs to a univariate KW distribution with two shape parameters $\alpha $ and $\theta +\delta .$ The parameter $\beta $ represents the scale variation from a standard KW model.

#### 2.2. Conditional Distributions

`Mathematica`. Similarly, one can find an expression for the conditional density of X given $Y=y$ and the corresponding conditional moments.

#### 2.3. Marginal Moments

`Mathematica`), from the marginal density of Y from Equation (8), we have

#### 2.4. Product Moment

`Mathematica`.

**Corollary 1.**

**Proof.**

#### 2.5. Distributional Properties

- The bivariate density in Equation (1) is positive regression-dependent (PRD) (i.e.,$P\left(X\le x|Y=y\right)$ is decreasing in y for all $x,$ and similarly, $P\left(Y\le y|X=x\right)$ is decreasing in x for all y).
- Furthermore, the property of being PRD will imply that $P\left(Y\le y|X\le x\right)$ is non-decreasing in x for all y and that $P\left(Y\le y|X\le x\right)$ is non-increasing in x for all $y,$ each of which imply that $P\left(Y>y|X>x\right)\ge P\left(Y>y\right)P\left(X>x\right)$ and $P\left(Y\le y|X\le x\right)\ge P\left(Y\le y\right)P\left(X\le x\right)$, namely such that X and Y are positive quadrant-dependent (PQD).

## 3. Model 2’s Structural Properties

**Marginal densities:**

- Observe that as $x\to 0,$$f\left(x,y\right)\sim C\alpha \beta {\theta}_{1}{\theta}_{2}{\gamma}_{1}{\gamma}_{2}{x}^{\alpha -1}{y}^{\beta -1}{\left(1-{\theta}_{2}{y}^{\beta}\right)}^{\delta +{\gamma}_{2}-\alpha -\beta -1}.$ Thus, the distribution of Y along the vertical boundary of the simplex belongs to a univariate KW distribution with two shape parameters $\beta $ and $\delta +{\gamma}_{2}-\alpha -\beta -1.$ The parameter ${\theta}_{2}$ represents the scale variation from a standard KW model.
- Likewise, as $y\to 0,$$f\left(x,y\right)\sim C\alpha \beta {\theta}_{1}{\theta}_{2}{\gamma}_{1}{\gamma}_{2}{x}^{\alpha -1}{y}^{\beta -1}{\left(1-{\theta}_{1}{x}^{\alpha}\right)}^{\delta +{\gamma}_{1}-\alpha -\beta -1}.$ Thus, the distribution of Y along the vertical boundary of the simplex belongs to a univariate KW distribution with two shape parameters $\beta $ and $\delta +{\gamma}_{1}-\alpha -\beta -1.$ The parameter ${\theta}_{1}$ represents the scale variation from a standard KW model.

#### Distributional Properties

## 4. Inference

#### 4.1. Method of Moments Estimation

#### 4.2. Maximum Likelihood Estimation

## 5. Simulation

- Generate independent Kumaraswamy random variables U and V with shape parameters $\left(\alpha ,\delta \right)$ and $\left(\beta ,\delta \right),$ respectively.
- Generate a uniform [0, 1] random variable W independent from $\left(U,V\right).$
- If $W<D{\left({\left(1-\theta \right)}^{\delta}\left[{\phantom{(}}_{2}{F}_{1}\left(1,\theta +\delta ;1+\theta +\delta ;\frac{\theta}{(1-\theta {\left(UV\right)}^{\alpha})}\right)\right]\right)}^{-1}$, then accept $\left(U,V\right)$ as a realization from the bivariate density in Equation (1).
- If $W\ge D{\left({\left(1-\theta \right)}^{\delta}\left[{\phantom{(}}_{2}{F}_{1}\left(1,\theta +\delta ;1+\theta +\delta ;\frac{\theta}{(1-\theta {\left(UV\right)}^{\alpha})}\right)\right]\right)}^{-1},$ then return to step $1.$

**Remark 1.**

## 6. Real Data Application

- Data Set I: Earthquakes become major societal risks when they strike vulnerable populations. We consider the data obtained from [7]. Due to the fact that a significant portion of Turkey is subject to frequent earthquakes, destructive mainshocks and their foreshock and aftershock sequences, the area between the longitudes 39 and 42${}^{\circ}$ N and latitudes 26 and 45${}^{\circ}$ E was investigated. In this particular region, 111 mainshocks with surface magnitudes (${M}_{s}$) of five or more occurred in the past 106 years. We define the following random variables. X represents the magnitude of the foreshocks, and Y represents the magnitude of the aftershocks. We fit the data to the following bivariate KW models.
- Data Set II: The data on 45 patients were available from a private clinic in Tennessee regarding the hemoglobin content in blood being prone to type II diabetes. To see the effect of reducing the hemoglobin content in the blood, a special type of treatment was administered to those patients. We define the following variables. X is a random variable which represents the proportion of hemoglobin content in the blood before the treatment, and Y is a random variable which represents the proportion of hemoglobin content in the blood after treatment.

- Model I: Bivariate distribution as defined in Equation (1);
- Model II: Bivariate distribution as defined in Equation (2);
- Model IV: Bivariate Kumaraswamy distribution via conditional specification, according to [5], given by$$\begin{array}{ccc}\hfill f\left(x,y\right)& =& C{\alpha}_{1}{\alpha}_{2}{x}^{{\alpha}_{1}-1}{y}^{{\alpha}_{2}-1}{(1-{x}^{{\alpha}_{1}})}^{{\beta}_{1}-1}{(1-{y}^{{\alpha}_{2}})}^{{\beta}_{2}-1}exp\left({\beta}_{3}log(1-{x}^{{\alpha}_{1}})log(1-{x}^{{\alpha}_{2}})\right)\hfill \\ & & \times I\left(0<x<1,0<y<1\right),\hfill \end{array}$$
- Model V: Bivariate Kumaraswamy distribution via conditional survival specification, according to Arnold and Ghosh (2016), given by$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& f\left(x,y\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& ={\alpha}_{1}{\alpha}_{2}{x}_{1}^{{\alpha}_{1}-1}{y}^{{\alpha}_{2}-1}{(1-{x}^{{\alpha}_{1}})}^{{\beta}_{1}-1}{(1-{y}^{{\alpha}_{2}})}^{{\beta}_{2}-1}exp{\beta}_{3}log(1-{x}_{1}^{{\alpha}_{1}})log(1-{y}^{{\alpha}_{2}})\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \times \left({\beta}_{1}{\beta}_{2}+{\beta}_{2}{\beta}_{3}log(1-{x}_{1}^{{\alpha}_{1}})+{\beta}_{3}+{\beta}_{3}^{2}log(1-{x}^{{\alpha}_{1}})log(1-{y}^{{\alpha}_{2}})+{\beta}_{1}{\beta}_{3}log(1-{y}^{{\alpha}_{2}})\right)I(0<{x}_{1},{x}_{2}<1).\hfill \end{array}$$
- Model VI: [11] bivariate ${F}_{3}$’s beta distribution, given by$$f\left(x,y\right)=\frac{C{x}^{\beta -1}{y}^{\delta -1}{\left(1-x-y\right)}^{\gamma -\beta -\delta -1}}{{\left(1-ux\right)}^{{\theta}_{1}}{\left(1-vy\right)}^{{\theta}_{2}}},$$
- Model VII: [12] bivariate generalized beta distribution given by$$f\left(x,y\right)=\frac{C{x}^{\alpha -1}{y}^{\beta -1}{\left(1-x\right)}^{\gamma -\alpha -1}{\left(1-y\right)}^{\gamma -\beta -1}}{{\left(1-xy\delta \right)}^{\gamma}},$$

`constrOptim`in $R.$ In addition, regarding fitting of the marginal densities of X and Y, as suggested by one reviewer, we report the following goodness of summary statistics for the bivariate probability model 1 in Equation (1) below. Furthermore, we have also included the bivariate scatterplot for the first data set along with the graphs related to marginal density plots in Appendix A.

- Here are the results for the K-S goodness of fit for Data Set I:
- For the marginal density of X, K-S value = 0.0648 and K-S p-value = 0.7946.
- For the marginal density of Y, the K-S value = 0.06893 and K-S p-value = 0.8139.

- Here are the results for the K-S goodness of fit for Data Set II:
- For the marginal density of $X,$ the K-S value = 0.06794 and K-S p-value = 0.8233.
- For the marginal density of $Y,$ the K-S value = 0.06853 and K-S p-value = 0.8394.

## 7. Concluding Remarks

- $$f\left({\underline{x}}^{p\times 1}\right)={M}_{1}\prod _{i=1}^{p}{\alpha}_{i}{x}_{i}^{{\alpha}_{i}-1}{\left(1-{x}_{i}^{{\alpha}_{i}}\right)}^{\theta +{\delta}_{1}}{\left(1-\theta {x}_{i}^{{\alpha}_{i}}\right)}^{\delta}I\left(0<{x}_{i}<1\right),$$
- Another such model could be$$f\left({\underline{x}}^{p\times 1}\right)={M}_{2}\prod _{i=1}^{p}{\theta}_{i}{x}_{i}^{{\alpha}_{i}-1}{\left(1-{\theta}_{i}{x}_{i}^{{\alpha}_{i}}\right)}^{\delta -{\alpha}_{i}-1}I\left(0<{x}_{i}<1\right),$$

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Figure A1.**The bivariate Kumaraswamy density (Equation (1)) plot, where $\alpha =0.5,\beta =0.5,\theta =0.5$ and $\delta =2.$

**Figure A2.**The bivariate Kumaraswamy contour (Equation (1)) plot, where $\alpha =0.5,\beta =0.5,\theta =0.5$ and $\delta =2.$

**Figure A3.**The bivariate Kumaraswamy density (Equation (1)) plot, where $\alpha =0.8,\beta =0.8,\theta =0.2$ and $\delta =3.$

**Figure A4.**The bivariate Kumaraswamy contour (Equation (1)) plot, where $\alpha =0.8,\beta =0.8,\theta =0.2$ and $\delta =3.$

**Figure A5.**The bivariate Kumaraswamy density (Equation (1)) plot, where $\alpha =0.8,\beta =0.8,\theta =0.4$ and $\delta =3.$

**Figure A6.**The bivariate Kumaraswamy contour (Equation (1)) plot, where $\alpha =0.8,\beta =0.8,\theta =0.4$ and $\delta =3.$

**Figure A7.**The bivariate Kumaraswamy density (Equation (1)) plot, where $\alpha =0.5,\beta =0.8,\theta =1.2$ and $\delta =3.$

**Figure A8.**The bivariate Kumaraswamy contour (Equation (1)) plot, where $\alpha =0.8,\beta =0.8,\theta =1.2$ and $\delta =3.$

**Figure A9.**The bivariate Kumaraswamy density (Equation (1)) plot, where $\alpha =0.8,\beta =0.8,\theta =4.5$ and $\delta =3.$

**Figure A10.**The bivariate Kumaraswamy contour (Equation (1)) plot, where $\alpha =0.8,\beta =0.8,\theta =4.5$ and $\delta =3.$

**Figure A11.**The bivariate Kumaraswamy density (Equation (1)) plot, where $\alpha =1.5,\beta =2.5,\theta =0.5$ and $\delta =0.9.$

**Figure A12.**The bivariate Kumaraswamy contour (Equation (1)) plot, where $\alpha =1.5,\beta =2.5,\theta =0.5$ and $\delta =0.9.$

**Figure A13.**The bivariate Kumaraswamy density (Equation (1)) plot, where $\alpha =2.5,\beta =1.5,\theta =0.5$ and $\delta =0.9.$

**Figure A14.**The bivariate Kumaraswamy contour (Equation (1)) plot, where $\alpha =2.5,\beta =1.5,\theta =0.5$ and $\delta =0.9.$

**Figure A15.**The bivariate Kumaraswamy density (Equation (1)) plot, where $\alpha =1.1,\beta =1.1,\theta =0.8$ and $\delta =2.$

**Figure A16.**The bivariate Kumaraswamy contour (Equation (1)) plot, where $\alpha =1.1,\beta =1.1,\theta =0.8$ and $\delta =2.$

**Figure A17.**The bivariate Kumaraswamy density (Equation (1)) plot, where $\alpha =1.1,\beta =1.1,\theta =2$ and $\delta =0.8.$

**Figure A18.**The bivariate Kumaraswamy contour (Equation (1)) plot, where $\alpha =1.1,\beta =1.1,\theta =2$ and $\delta =0.8.$

**Figure A20.**The bivariate KW marginal densities (Equation (1)) plot for the first data set.

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Parameter | Sample Size ($\mathit{n}=50$) | Sample Size ($\mathit{n}=100$) | Sample Size ($\mathit{n}=300$) |
---|---|---|---|

$\alpha $ | 0.1211 (0.3548) | 0.0769 (0.2271) | 0.0451 (0.1718) |

$\beta $ | −0.0148 (0.03748) | 0.0132 (0.02616) | 0.00118 (0.0135) |

$\theta $ | 0.0962 (0.0452) | 0.0473 (0.0224) | 0.00945 (0.1154) |

$\delta $ | 0.1452 (0.1094) | 0.1317 (0.0248) | 0.00438 (0.0176) |

Model | Model I | Model II | Model III | Model IV |
---|---|---|---|---|

Parameter estimates | $\widehat{\alpha}=2.1341\phantom{\rule{3.33333pt}{0ex}}\left(1.3214\right)$ | $\widehat{\alpha}=2.432\phantom{\rule{3.33333pt}{0ex}}\left(0.6853\right)$ | $\widehat{{\alpha}_{1}}=1.1849\phantom{\rule{3.33333pt}{0ex}}\left(0.6402\right)$ | $\widehat{{\alpha}_{1}}=2.2981\phantom{\rule{3.33333pt}{0ex}}\left(0.5234\right)$ |

$\widehat{\beta}=1.8453\phantom{\rule{3.33333pt}{0ex}}\left(1.381\right)$ | $\widehat{\beta}=1.5422\phantom{\rule{3.33333pt}{0ex}}\left(0.6732\right)$ | $\widehat{{\alpha}_{2}}=1.1538\phantom{\rule{3.33333pt}{0ex}}\left(0.3614\right)$ | $\widehat{{\alpha}_{2}}=2.1478\phantom{\rule{3.33333pt}{0ex}}\left(0.3427\right)$ | |

$\widehat{\theta}=0.3165\phantom{\rule{3.33333pt}{0ex}}\left(0.0842\right)$ | $\widehat{{\gamma}_{1}}=2.2428\phantom{\rule{3.33333pt}{0ex}}\left(0.1065\right)$ | $\widehat{{\beta}_{1}}=1.472\phantom{\rule{3.33333pt}{0ex}}\left(0.1368\right)$ | $\widehat{{\beta}_{1}}=2.1138\phantom{\rule{3.33333pt}{0ex}}\left(0.8436\right)$ | |

$\widehat{\delta}=0.4576\phantom{\rule{3.33333pt}{0ex}}\left(0.1248\right)$ | $\widehat{{\gamma}_{2}}=1.432\phantom{\rule{3.33333pt}{0ex}}\left(1.3842\right)$ | $\widehat{{\beta}_{2}}=1.5329\phantom{\rule{3.33333pt}{0ex}}\left(0.2829\right)$ | $\widehat{{\beta}_{2}}=2.1769\phantom{\rule{3.33333pt}{0ex}}\left(1.2283\right)$ | |

$\widehat{{\theta}_{1}}=0.7982\phantom{\rule{3.33333pt}{0ex}}\left(0.6425\right)$ | $\widehat{{\beta}_{3}}=1.6825\phantom{\rule{3.33333pt}{0ex}}\left(0.6297\right)$ | $\widehat{{\beta}_{3}}=1.457\phantom{\rule{3.33333pt}{0ex}}\left(0.4126\right)$ | ||

$\widehat{{\theta}_{2}}=1.0317\phantom{\rule{3.33333pt}{0ex}}\left(2.3845\right)$ | ||||

Log likelihood | −234.18 | −242.67 | −265.46 | −274.83 |

${\chi}^{2}$ value | 0.1485 | 0.3518 | 1.0678 | 2.1019 |

AIC | 476.36 | 497.34 | 540.92 | 559.66 |

BIC | 472.17 | 489.15 | 534.73 | 553.466 |

Model | Model I | Model II | Model III | Model IV |
---|---|---|---|---|

Parameter estimates | $\widehat{\alpha}=3.2849\phantom{\rule{3.33333pt}{0ex}}\left(1.0872\right)$ | $\widehat{\alpha}=2.2578\phantom{\rule{3.33333pt}{0ex}}\left(0.5241\right)$ | $\widehat{{\alpha}_{1}}=2.3244\phantom{\rule{3.33333pt}{0ex}}\left(0.6402\right)$ | $\widehat{{\alpha}_{1}}=1.15681\phantom{\rule{3.33333pt}{0ex}}\left(0.5423\right)$ |

$\widehat{\beta}=1.4546\phantom{\rule{3.33333pt}{0ex}}\left(0.3371\right)$ | $\widehat{\beta}=1.2829\phantom{\rule{3.33333pt}{0ex}}\left(0.9562\right)$ | $\widehat{{\alpha}_{2}}=2.0172\phantom{\rule{3.33333pt}{0ex}}\left(0.4718\right)$ | $\widehat{{\alpha}_{2}}=2.1342\phantom{\rule{3.33333pt}{0ex}}\left(0.5819\right)$ | |

$\widehat{\theta}=0.4263\phantom{\rule{3.33333pt}{0ex}}\left(0.1842\right)$ | $\widehat{{\gamma}_{1}}=2.2483\phantom{\rule{3.33333pt}{0ex}}\left(0.4061\right)$ | $\widehat{{\beta}_{1}}=1.5233\phantom{\rule{3.33333pt}{0ex}}\left(0.5319\right)$ | $\widehat{{\beta}_{1}}=3.1826\phantom{\rule{3.33333pt}{0ex}}\left(0.5941\right)$ | |

$\widehat{\delta}=0.5843\phantom{\rule{3.33333pt}{0ex}}\left(0.1268\right)$ | $\widehat{{\gamma}_{2}}=1.923\phantom{\rule{3.33333pt}{0ex}}\left(0.9537\right)$ | $\widehat{{\beta}_{2}}=1.1883\phantom{\rule{3.33333pt}{0ex}}\left(0.2445\right)$ | $\widehat{{\beta}_{2}}=2.16782\phantom{\rule{3.33333pt}{0ex}}\left(0.2156\right)$ | |

$\widehat{{\theta}_{1}}=1.2061\phantom{\rule{3.33333pt}{0ex}}\left(2.2346\right)$ | $\widehat{{\beta}_{3}}=1.6542\phantom{\rule{3.33333pt}{0ex}}\left(0.3273\right)$ | $\widehat{{\beta}_{3}}=2.6381\phantom{\rule{3.33333pt}{0ex}}\left(0.4018\right)$ | ||

$\widehat{{\theta}_{2}}=0.8924\phantom{\rule{3.33333pt}{0ex}}\left(1.0617\right)$ | ||||

Log likelihood | −212.17 | −216.32 | −238.49 | −255.38 |

${\chi}^{2}$ value | 0.1378 | 0.1416 | 0.5516 | 0.5492 |

AIC | 432.34 | 444.64 | 486.98 | 520.76 |

BIC | 429.049 | 437.35 | 481.68 | 515.469 |

Model | Model V | Model VI | Model VII |
---|---|---|---|

Parameter estimates | $\widehat{{\alpha}_{1}}=1.4723\phantom{\rule{3.33333pt}{0ex}}\left(0.5582\right)$ | $\widehat{\gamma}=0.6487\phantom{\rule{3.33333pt}{0ex}}\left(0.6673\right)$ | $\widehat{\alpha}=2.1162\phantom{\rule{3.33333pt}{0ex}}\left(0.5436\right)$ |

$\widehat{{\alpha}_{2}}=0.9503\phantom{\rule{3.33333pt}{0ex}}\left(1.462\right)$ | $\widehat{\beta}=1.773\phantom{\rule{3.33333pt}{0ex}}\left(0.4249\right)$ | $\widehat{\beta}=2.5648\phantom{\rule{3.33333pt}{0ex}}\left(1.2838\right)$ | |

$\widehat{{\beta}_{1}}=1.0892\phantom{\rule{3.33333pt}{0ex}}\left(4.5253\right)$ | $\widehat{\delta}=4.229\phantom{\rule{3.33333pt}{0ex}}\left(1.3835\right)$ | $\widehat{\gamma}=3.182\phantom{\rule{3.33333pt}{0ex}}\left(0.2896\right)$ | |

$\widehat{{\beta}_{2}}=1.6534\phantom{\rule{3.33333pt}{0ex}}\left(1.2435\right)$ | $\widehat{{\theta}_{1}}=1.4064\phantom{\rule{3.33333pt}{0ex}}\left(1.1529\right)$ | $\widehat{\delta}=1.0729\phantom{\rule{3.33333pt}{0ex}}\left(1.3826\right)$ | |

$\widehat{{\beta}_{3}}=2.0308\phantom{\rule{3.33333pt}{0ex}}\left(0.8425\right)$ | $\widehat{{\theta}_{2}}=1.4592\phantom{\rule{3.33333pt}{0ex}}\left(0.6839\right)$ | ||

Log likelihood | −245.48 | −267.18 | −286.65 |

${\chi}^{2}$ value | 0.5237 | 0.6225 | 0.7244 |

AIC | 501.16 | 544.36 | 581.30 |

BIC | 494.77 | 538.17 | 577.11 |

Model | Model V | Model VI | Model VII |
---|---|---|---|

Parameter estimates | $\widehat{\beta}=1.893\phantom{\rule{3.33333pt}{0ex}}\left(0.7532\right)$ | $\widehat{\alpha}=2.4325\phantom{\rule{3.33333pt}{0ex}}\left(1.2485\right)$ | $\widehat{\alpha}=2.4573\phantom{\rule{3.33333pt}{0ex}}\left(0.8771\right)$ |

$\widehat{{\alpha}_{2}}=0.8845\phantom{\rule{3.33333pt}{0ex}}\left(1.4468\right)$ | $\widehat{\beta}=1.7562\phantom{\rule{3.33333pt}{0ex}}\left(0.6248\right)$ | $\widehat{\beta}=2.5648\phantom{\rule{3.33333pt}{0ex}}\left(1.2835\right)$ | |

$\widehat{{\beta}_{1}}=1.0892\phantom{\rule{3.33333pt}{0ex}}\left(2.2551\right)$ | $\widehat{\delta}=2.4326\phantom{\rule{3.33333pt}{0ex}}\left(1.4542\right)$ | $\widehat{\gamma}=1.0123\phantom{\rule{3.33333pt}{0ex}}\left(0.6842\right)$ | |

$\widehat{{\beta}_{2}}=1.5422\phantom{\rule{3.33333pt}{0ex}}\left(0.7494\right)$ | $\widehat{{\theta}_{1}}=1.5645\phantom{\rule{3.33333pt}{0ex}}\left(0.9326\right)$ | $\widehat{\delta}=2.0106\phantom{\rule{3.33333pt}{0ex}}\left(1.2363\right)$ | |

$\widehat{{\beta}_{3}}=1.8634\phantom{\rule{3.33333pt}{0ex}}\left(0.5438\right)$ | $\widehat{{\theta}_{2}}=1.1672\phantom{\rule{3.33333pt}{0ex}}\left(1.2648\right)$ | ||

Log likelihood | −262.17 | −258.34 | −247.38 |

${\chi}^{2}$ value | 0.5632 | 0.4785 | 0.3451 |

AIC | 534.34 | 526.68 | 502.76 |

BIC | 529.049 | 521.38 | 499.47 |

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## Share and Cite

**MDPI and ACS Style**

Ghosh, I.
A New Class of Alternative Bivariate Kumaraswamy-Type Models: Properties and Applications. *Stats* **2023**, *6*, 232-252.
https://doi.org/10.3390/stats6010014

**AMA Style**

Ghosh I.
A New Class of Alternative Bivariate Kumaraswamy-Type Models: Properties and Applications. *Stats*. 2023; 6(1):232-252.
https://doi.org/10.3390/stats6010014

**Chicago/Turabian Style**

Ghosh, Indranil.
2023. "A New Class of Alternative Bivariate Kumaraswamy-Type Models: Properties and Applications" *Stats* 6, no. 1: 232-252.
https://doi.org/10.3390/stats6010014