# A New Kumaraswamy Generalized Family of Distributions with Properties, Applications, and Bivariate Extension

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## Abstract

**:**

## 1. Introduction

**Note 1.**The citations and the references of the authors of special models of the Kw-G family [8] are avoided in this section and in references to save some space.

- (i)
- $W\left[G\right(x\left)\right]\in [a,b]$,
- (ii)
- $W\left[G\right(x\left)\right]$ is differentiable and monotonically non-decreasing, and
- (iii)
- $\underset{x\to -\infty}{lim}W\left[G\left(x\right)\right]=a$ and $\underset{x\to \infty}{lim}W\left[G\left(x\right)\right]=b$.

- (i)
- Constructing new and novel G-families as a function of a cdf, $W\left[G\right(x\left)\right]$, is a difficult task in these days. A few pioneer G-families were developed in the literature considering $W\left[G\right(x\left)\right]$ viz. exponentiated-G with power parameter $\alpha >0$ (LA1 and LA2) (Gupta et al. [3]) [$G{\left(x\right)}^{\alpha}\phantom{\rule{0.166667em}{0ex}}\mathrm{and}\phantom{\rule{0.166667em}{0ex}}1-\overline{G}{\left(x\right)}^{\alpha}$], beta-G (Eugene et al. [4]) [$G\left(x\right)$], ZBgamma-G (Zografos and Balakrishnan [7]) [$-log\overline{G}\left(x\right)$], odd log-logistic-G (Gleaton and Lynch [5]) [$G\left(x\right)/\overline{G}\left(x\right)$], RBgamma-G (Ristic and Balakrishnan [10]) [$-logG\left(x\right)$], log-odd logistic-G (Torabi and Montazeri [22]) [$log\{G\left(x\right)/\overline{G}\left(x\right)\}$], Gumbel-X (Al-Aqtash et al. [23]) [$log\{-log\phantom{\rule{0.166667em}{0ex}}\overline{G}\left(x\right)\}$], Weibull-X (T-X approach) and Weibull-X (Ahmad et al. [24]) [$\{-log\phantom{\rule{0.166667em}{0ex}}\overline{G}\left(x\right)\}/\overline{G}\left(x\right)$] are the pioneer works. Other G-families either non-composite (alone based on well-established parent model) or composite (mixture of two G-families) and compounded G-families are the extensions or modifications of the above described pioneer G-families. For example, the generator $G\left(x\right)$, where $T\in (0,1)$ was pioneered by (Eugene et al. [4]) for defining the beta-G family, and later this generator was adopted by (Cordeiro and de-Castro [8]; Alexander et al. [9]; Rezaei et al. [15]) for defining the Kw-G, Mc-G and TL-G families, respectively. Similarly, the odd generator $G\left(x\right)/\overline{G}\left(x\right)$ (where $T\in (0,\infty )$) was suggested by (Gleaton and Lynch, [5]) for proposing the odd log-logistic-G family, and it was adopted by (Bourguignon et al. [12]; Torabi and Montazeri [25]; Tahir et al. [13]; Silva et al. [26]; Cordeiro et al. [27]; Alizadeh et al. [28]; Cordeiro et al. [29]; Hassan et al. [30]; Hassan and Nassr [31]; Maiti and Pramanik [32]); El-Morshedy and Eliwa [33], Alizadeh et al. [34]; El-Morshedy et al. [35]; Eliwa et al. [36] for defining the odd Weibull-G, odd gamma-G, odd generalized-exponential-G, odd Lindley-G, odd Burr-G, odd power-Cauchy-G, odd half-Cauchy, odd additive Weibull-G, odd power-Lindley-G, odd Xgamma-G, odd flexible Weibull-H, odd log-logistic Lindley-G, odd Chen generator and exponentiated odd Chen-G, respectively, among others.
- (ii)
- The proposed extension of the Kumaraswamy-G model is based on a new generator $W\left[G\left(x\right)\right]=1-\overline{G}{\left(x\right)}^{G\left(x\right)}$ for $T\in (0,1)$ instead of the only existing generator $G\left(x\right)$ for which the beta-G, Kw-G, Mc-G and TL-G classes were developed so far.
- (iii)
- The proposed generator $1-\overline{G}{\left(x\right)}^{G\left(x\right)}$ seems little complicated in comparison to earlier well-established generator for the unit interval but it has the ability to produce better estimates and goodness-of-fit (GoF) tests results that can make it distinguishable and attractive for applied researchers (as evident from the results in Section 5 and Section 7).
- (iv)
- For most of the families and models, if the cdf is in closed form, then the quantile function (qf) can be straightforward to obtain. In some families and models, where the qf is based on some special functions such as beta, gamma, and others, then the qfs can only be determined by using power series. In our case, the cdf of the family is in closed form but the qf can be obtained only numerically.

**Note 2.**A complete and independent investigation of the properties and application of our proposed generator $F\left(x\right)=1-\overline{G}{\left(x\right)}^{G\left(x\right)}$ as a new family such as transmuted-G (Tr-G) and exponentiated-generalized-G (EG-G) will appear in another outlet very soon. It is noted here that the two G-families (Tr-G and EG-G) have not been developed from any existing parent model similar to our proposed one.

## 2. The NKw-G Family

## 3. Properties of the NKw-G Family

#### 3.1. Quantile Function

- (i)
- Set $z=z\left(u\right)=1-{[1-{(1-u)}^{1/b}]}^{1/a}$;
- (ii)
- Find $w=w\left(u\right)$ numerically in $wlog(1-w)=log\left(z\right)$ using any Newton-Raphson algorithm;
- (iii)
- Solving numerically for x in $G(x;\xi )=w$ yields the qf $x=Q\left(u\right)$ of X.

#### 3.2. Asymptotics

**Corollary**

**1.**

**Corollary**

**2.**

#### 3.3. Analytic Shapes of the Density and Hazard Rate Function

#### 3.4. Linear Representation of the NKw-G Density

#### 3.5. Mathematical Properties

#### 3.6. Estimation

`R`program (

`optim`function),

`SAS`(

`PROC NLMIXED`),

`Ox`(sub-routine

`MaxBFGS`), among others.

`R`statistical computing environment (https://www.r-project.org/). An important advantage of this package is that it is not necessary to define the log-likelihood function and that it computes the MLEs, their standard errors and some GoF statistics. We only need to provide the pdf and cdf of the distribution to be fitted to a data set.

## 4. The NKwW Distribution

#### 4.1. Linear Representation

#### 4.2. Properties

#### 4.3. Quantile Function and Simulation Study

- Set n, $\alpha $, $\beta $, a, b and initial value ${x}^{0}$.
- Generate U ∼Uniform$(0,1)$.
- Update ${x}^{0}$ by using the Newton’s formula$$x*={x}^{0}-R({x}^{0};\alpha ,\beta ,a,b),$$
- If $|{x}^{0}-{x}^{*}|\le \u03f5$, ($\u03f5>0$, very small tolerance limit), then store ${x}^{0}={x}^{*}$ as a variate from the NKwW($\alpha ,\beta ,a,b$) distribution.
- If $|{x}^{0}-{x}^{*}|>\u03f5$, then, set ${x}^{0}={x}^{*}$ and go to step 3.
- Repeat steps (2)–(5) n times to generate ${x}_{1},\cdots {x}_{n}$.

`R`script to generate observations from the NKwW distribution is given in the Appendix A.

#### 4.4. Estimation

## 5. Empirical Illustrations of NKwW Model

`R`. The unknown parameters of the models are estimated by the maximum likelihood method. The log-likelihood function is evaluated at the MLEs ($\widehat{\ell}$). The well-known GoFS such as the Akaike information criterion (AIC), Bayesian Information Criterion (BIC), Hannan-Quinn Information Criterion (HQIC), Anderson-Darling (${A}^{*}$), Cramér–von Mises (${W}^{*}$) and Kolmogrov-Smirnov (K-S) are adopted for model comparisons. The lower values of GoFS and higher p-values of the K-S statistic indicate good fits.

## 6. Bivariate New Kumaraswamy G-Family

#### The MLE for the BvNKw-G Family

## 7. Empirical Illustrations of BvNKwW Model Through Motors Data

## 8. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

`R`script to generate NKwW variates is given below:

`n=20; alpha=1; beta=1.5; a=2.5;b=2.5;`

`f=function(x,alpha,beta,a,b)`

`{`

`g=alpha*beta*x^(beta-1))*exp(-alpha*x^{beta})`

`G=1-exp(-alpha*x^{beta})`

`F=1-(1-(1-(1-G)^G)^a)^b`

`D =a*b*g*(1-G)^G*(1-(1-G)^G)^(a-1))`

`*((G)/(1-G)-log(1-G))*(1-(1-(1-G)^G)^a)^(b-1)`

`return(D)`

`};`

`F=function(x,alpha,beta,a,b)`

`{`

`g=alpha*beta*x^(beta-1))*exp(-alpha* x^{beta})`

`G=1-exp(-alpha*x^{beta})`

`F=1-(1-(1-(1-G)^G)^a)^b`

`D =a*b*g*(1-G)^G*(1-(1-G)^G)^(a-1))*((G)/(1-G)-log(1-G))`

`*(1-(1-(1-G)^G)^a)^(b-1)`

`return(d)`

`};`

`u=runif(n,0,1);`

`x=rep(0,n);`

`for(i in 1:n)`

`{`

`x0=1`

`xnew=x0-((F(x0,alpha,beta,a,b)-u[i])/f(x0,alpha,beta,a,b))`

`while(abs(xnew-x0) > 0.0001)`

`{`

`x0=xnew`

`xnew=x0-((F(x0,alpha,beta,a,b)-u[i])/f(x0,alpha,beta,a,b))`

`}`

`x[i]=xnew`

`}`

`print(x)`

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**Figure 3.**Skewness and kurtosis plots of the NKwW model for some values of $\alpha $ = 1.5, 0.5, 1.5 and $\beta $ = 2.5, 3.5, 0.5.

**Figure 8.**The surface plots of the jpdf, jhrf and jrf of the BvNKwW model for $a=0.6,{b}_{1}=4,{b}_{2}=4$, ${b}_{3}=4,\alpha =0.6$ and $\beta =2.9$.

**Figure 9.**The surface plots of the jpdf, jhrf and jrf of the BvNKwW model for $a=1.6,{b}_{1}=2,{b}_{2}=2$, ${b}_{3}=2,\alpha =1.6$ and $\beta =2.9$.

**Figure 10.**The surface plots of the jpdf, jhrf and jrf of the BvNKwW model for $a=0.8,{b}_{1}=1.5,{b}_{2}=1.5$, ${b}_{3}=1.5,\alpha =0.9$ and $\beta =1.9$.

Distribution | $\mathit{\alpha}$ | $\mathit{\beta}$ | a | b | $\mathit{\theta}$ |
---|---|---|---|---|---|

NKwW | 0.0089 | 1.1514 | 5.0192 | 0.5054 | – |

(0.0023) | (0.0730) | (1.6716) | (0.1807) | – | |

KwW | 0.0160 | 1.3962 | 5.7590 | 0.3381 | – |

(0.0027) | (0.2113) | (1.9264) | (0.1554) | – | |

BW | 0.0219 | 0.7969 | 13.2183 | 1.2340 | – |

(0.0076) | (0.1903) | (5.1706) | (0.9787) | – | |

EGW | 0.0086 | 0.9045 | 2.0898 | 10.5512 | – |

(0.0022) | (0.1288) | (0.6257) | (4.4416) | – | |

McW | 0.0132 | 1.2859 | 1.7925 | 0.5887 | 2.5824 |

(0.0032) | (0.2544) | (0.6126) | (0.3831) | (0.9474) | |

GaW | 1.1306 | 0.5469 | 17.6158 | – | – |

(0.0496) | (0.0154) | (1.4970) | – | – | |

OLLW | 0.0045 | 1.0313 | 2.4836 | – | – |

(0.0003) | (0.1759) | (0.4532) | – | – | |

MOW | 0.0032 | 3.4739 | – | – | 0.1039 |

(0.0002) | (0.3436) | – | – | (0.0489) | |

TrW | 0.0043 | 2.4549 | – | – | 0.6144 |

(0.0003) | (0.1755) | – | – | (0.2078) | |

W | 0.0050 | 2.2745 | – | – | – |

(0.0002) | (0.1629) | – | – | – |

Distribution | $\mathit{\alpha}$ | $\mathit{\beta}$ | a | b | $\mathit{\theta}$ |
---|---|---|---|---|---|

NKwW | 0.1742 | 0.9887 | 59.0160 | 0.2183 | – |

(0.0316) | (0.0619) | (0.4024) | (0.0585) | – | |

KwW | 0.1609 | 1.0252 | 54.7825 | 0.2041 | – |

(0.0153) | (0.0276) | (0.1358) | (0.0382) | – | |

BW | 0.1320 | 1.1080 | 23.0602 | 0.1940 | – |

(0.0073) | (0.0068) | (8.7941) | (0.0324) | – | |

EGW | 0.0090 | 0.7774 | 5.5966 | 10.5493 | – |

(0.0041) | (0.1370) | (2.0458) | (5.6821) | – | |

McW | 0.1608 | 1.0049 | 14.5078 | 0.2210 | 2.5180 |

(0.0340) | (0.0466) | (9.8227) | (0.0757) | (0.0895) | |

GaW | 4.6144 | 0.4983 | 14.7225 | – | – |

(0.1518) | (0.0217) | (1.7239) | – | – | |

OLLW | 0.0154 | 0.9508 | 2.3925 | – | – |

(0.0021) | (0.3378) | (0.9487) | – | – | |

MOW | 0.0065 | 3.3556 | – | – | 0.0145 |

(0.0014) | (0.4292) | – | – | (0.0146) | |

TrW | 0.0137 | 1.9476 | – | – | 0.7003 |

(0.0016) | (0.1941) | – | – | (0.2483) | |

W | 0.0171 | 1.7719 | – | – | – |

(0.0015) | (0.1776) | – | – | – |

K-S | ||||||||
---|---|---|---|---|---|---|---|---|

Distribution | $\widehat{\mathit{\ell}}$ | AIC | BIC | HQIC | ${\mathit{A}}^{*}$ | ${\mathit{W}}^{*}$ | K-S | P-Value |

NKwW | 565.2337 | 1138.4670 | 1148.8880 | 1142.6850 | 0.1722 | 0.0207 | 0.0454 | 0.9863 |

KwW | 566.6253 | 1141.2510 | 1151.6710 | 1145.4680 | 0.3678 | 0.0477 | 0.0572 | 0.8987 |

BW | 566.2292 | 1140.4580 | 1150.8790 | 1144.6760 | 0.3149 | 0.0411 | 0.0489 | 0.9707 |

EGW | 566.2248 | 1140.4500 | 1150.8700 | 1144.6670 | 0.3266 | 0.0427 | 0.0487 | 0.9718 |

McW | 567.4362 | 1144.8720 | 1157.8980 | 1150.1440 | 0.4868 | 0.0655 | 0.0596 | 0.8695 |

GaW | 567.2618 | 1140.5240 | 1148.3390 | 1143.6870 | 0.5071 | 0.0689 | 0.0547 | 0.9257 |

OLLW | 569.6909 | 1145.3820 | 1153.1970 | 1148.5450 | 0.6649 | 0.0932 | 0.0807 | 0.5335 |

MOW | 568.4818 | 1142.9640 | 1150.7790 | 1146.1270 | 0.6431 | 0.0866 | 0.0595 | 0.8713 |

TrW | 573.7855 | 1153.5710 | 1161.3870 | 1156.7340 | 1.4659 | 0.2183 | 0.0872 | 0.4321 |

W | 576.1180 | 1156.2360 | 1161.4460 | 1158.3450 | 1.8275 | 0.2767 | 0.0936 | 0.3450 |

K-S | ||||||||
---|---|---|---|---|---|---|---|---|

Distribution | $\widehat{\mathit{\ell}}$ | AIC | BIC | HQIC | ${\mathit{A}}^{*}$ | ${\mathit{W}}^{*}$ | K-S | P-Value |

NKwW | 215.1742 | 438.3485 | 445.8333 | 441.1770 | 0.2003 | 0.0277 | 0.0776 | 0.9346 |

KwW | 215.5195 | 439.0389 | 446.5238 | 441.8675 | 0.2495 | 0.0347 | 0.0834 | 0.8924 |

BW | 216.1573 | 440.3147 | 447.7995 | 443.1432 | 0.3387 | 0.0477 | 0.0973 | 0.7538 |

EGW | 218.1801 | 444.3601 | 451.8449 | 447.1887 | 0.6147 | 0.0913 | 0.0973 | 0.7543 |

McW | 215.7566 | 441.5132 | 450.8692 | 445.0489 | 0.2699 | 0.0374 | 0.0837 | 0.8895 |

GaW | 219.4700 | 444.9401 | 450.5537 | 447.0615 | 0.8278 | 0.1250 | 0.1176 | 0.5203 |

OLLW | 220.4104 | 446.8208 | 452.4344 | 448.9422 | 0.9051 | 0.1388 | 0.0934 | 0.7966 |

MOW | 218.2594 | 442.5187 | 448.1323 | 444.6401 | 0.5773 | 0.0868 | 0.0791 | 0.9247 |

TrW | 224.0997 | 454.1994 | 459.8130 | 456.3208 | 1.5006 | 0.2372 | 0.1291 | 0.4001 |

W | 225.7065 | 455.4131 | 459.1555 | 456.8273 | 1.7286 | 0.2765 | 0.1399 | 0.3048 |

${\mathit{X}}_{1}$ | ${\mathit{X}}_{2}$ | $max({\mathit{X}}_{1},{\mathit{X}}_{2})$ | |||||||
---|---|---|---|---|---|---|---|---|---|

Model | −L | K-S | P-Value | −L | K-S | P-Value | −L | K-S | P-Value |

NKwW | $100.2890$ | $0.2376$ | $0.2614$ | $102.9142$ | $0.0902$ | $0.9956$ | $101.1965$ | $0.1372$ | $0.8871$ |

Model | |||||||
---|---|---|---|---|---|---|---|

Statistic | BvNKwW | BvGPW | BvEW | BvW | BvGEx | BvEx | BvGLFR |

$\widehat{a}$ | $1.6395$ | $0.0291$ | $0.5203$ | $0.0389$ | $0.0137$ | − | $6.99\times {10}^{-5}$ |

$\left(0.0651\right)$ | $\left(\right)$ | $\left(\right)$ | $\left(\right)$ | $\left(\right)$ | − | $\left(\right)$ | |

$\widehat{{b}_{1}}$ | $3.1333$ | 1.5591 | 30.1381 | 0.2004 | 2.4541 | 0.0023 | 0.4171 |

$\left(0.2364\right)$ | $\left(\right)$ | $\left(\right)$ | $\left(\right)$ | $\left(\right)$ | $\left(\right)$ | $\left(\right)$ | |

$\widehat{{b}_{2}}$ | $3.3989$ | $1.8581$ | $24.1351$ | $0.2383$ | $2.8803$ | $0.0021$ | $0.4864$ |

$\left(0.1896\right)$ | $\left(\right)$ | $\left(\right)$ | $\left(\right)$ | $\left(\right)$ | $\left(\right)$ | $\left(\right)$ | |

$\widehat{{b}_{3}}$ | $4.2869$ | $3.7191$ | $61.8051$ | $0.3381$ | $6.0641$ | $0.0051$ | $1.0188$ |

$\left(0.0985\right)$ | $\left(\right)$ | $\left(\right)$ | $\left(\right)$ | $\left(\right)$ | $\left(\right)$ | $\left(\right)$ | |

$\widehat{\alpha}$ | $0.0008$ | $0.0291$ | $0.5203$ | $0.0389$ | $0.0137$ | − | $6.99\times {10}^{-5}$ |

$\left(0.0001\right)$ | $\left(\right)$ | $\left(\right)$ | $\left(\right)$ | $\left(\right)$ | − | $\left(\right)$ | |

$\widehat{\beta}$ | $1.2532$ | − | − | − | − | − | − |

$\left(0.3421\right)$ | − | − | − | − | − | − | |

$-L$ | $211.1711$ | $431.7909$ | $339.2656$ | $422.9532$ | $335.2312$ | $355.7323$ | $331.7681$ |

AIC | $434.3422$ | $871.5818$ | $688.5312$ | $853.9064$ | $678.4624$ | $717.4646$ | $673.5362$ |

CAIC | $441.9786$ | $874.6587$ | $693.5312$ | $856.9833$ | $681.5393$ | $719.1789$ | $678.5362$ |

BIC | $439.6844$ | $875.14328$ | $692.9831$ | $857.4679$ | $682.0239$ | $720.1357$ | $677.9881$ |

HQIC | $435.0788$ | $872.0729$ | $689.1451$ | $854.3975$ | $678.9535$ | $717.8329$ | $674.1501$ |

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## Share and Cite

**MDPI and ACS Style**

Tahir, M.H.; Hussain, M.A.; Cordeiro, G.M.; El-Morshedy, M.; Eliwa, M.S.
A New Kumaraswamy Generalized Family of Distributions with Properties, Applications, and Bivariate Extension. *Mathematics* **2020**, *8*, 1989.
https://doi.org/10.3390/math8111989

**AMA Style**

Tahir MH, Hussain MA, Cordeiro GM, El-Morshedy M, Eliwa MS.
A New Kumaraswamy Generalized Family of Distributions with Properties, Applications, and Bivariate Extension. *Mathematics*. 2020; 8(11):1989.
https://doi.org/10.3390/math8111989

**Chicago/Turabian Style**

Tahir, Muhammad H., Muhammad Adnan Hussain, Gauss M. Cordeiro, M. El-Morshedy, and M. S. Eliwa.
2020. "A New Kumaraswamy Generalized Family of Distributions with Properties, Applications, and Bivariate Extension" *Mathematics* 8, no. 11: 1989.
https://doi.org/10.3390/math8111989