The bivariate distributions are useful in simultaneous modeling of two random variables. These distributions provide a way to model models. The bivariate families of distributions are not much widely explored and in this article a new family of bivariate distributions is proposed. The new family will extend the univariate transmuted family of distributions and will be helpful in modeling complex joint phenomenon. Statistical properties of the new family of distributions are explored which include marginal and conditional distributions, conditional moments, product and ratio moments, bivariate reliability and bivariate hazard rate functions. The maximum likelihood estimation (MLE) for parameters of the family is also carried out. The proposed bivariate family of distributions is studied for the Weibull baseline distributions giving rise to bivariate transmuted Weibull (BTW) distribution. The new bivariate transmuted Weibull distribution is explored in detail. Statistical properties of the new BTW distribution are studied which include the marginal and conditional distributions, product, ratio and conditional momenst. The hazard rate function of the BTW distribution is obtained. Parameter estimation of the BTW distribution is also done. Finally, real data application of the BTW distribution is given. It is observed that the proposed BTW distribution is a suitable fit for the data used.
Transmuted Distributions(T−X) family of distributionsbivariate transmuted family of distributionsWeibull distributionmaximum likelihood estimationIntroduction
The probability distributions are widely used in several areas of life. Certain situations arise where the standard probability models are not capable of capturing complex behavior of the data and hence some extensions are required. Numerous methods are available in literature to extend a univariate probability distribution by adding some new parameters. These developments provide more flexible statistical distributions by adding new parameters to the baseline distribution. A simple method to extend any baseline distribution is proposed by Shaw et al. [1] and is known as the transmuted family of distributions. The cumulative distribution function (cdf) of this family is
FX(x)=G(x)[1+λ−λG(x)],
where G(x) is the cdf of the baseline distribution and λ is the transmutation parameter such that λ∈[−1,1].
Various researches have extended the transmuted family of distributions by using the (T−X) family of distributions proposed by Alzaatreh et al. [2]; see for example the transmuted-G family of distributions [3], the Kumaraswamy transmuted-G family of distributions by Afify et al. [4], the beta transmuted-H family by Afify et al. [5] and the T-transmuted X family of distributions by Jayakumar et al. [6].
Different situations arise where the joint modeling of two variables is required and in such cases the bivariate distributions are required. The development of bivariate distributions from univariate marginals has been an area of interest. Several methodologies have been proposed to develop the bivariate distribution from given univariate marginals. A classical method has been proposed by Gumbel [7] which generated a bivariate distribution from given univariate marginals and is known as Gumbel family of distributions. The joint cdf of Gumbel bivariate distribution is given as
F(x,y)=G(x)G(y)[1+γ{1−G(x)}{1−G(y)}],
where G(x) and G(y) are any marginal cdf’s and γ is an association parameter. This family has been studied by various authors. For example, the bivariate Kumaraswamy distribution has been studied by Barreto-Souza et al. [8], the bivariate Pareto distributions by Sankaran et al. [9] among others. Sarabia et al. [10] has extended the Beta−G family of distributions to the bivariate case by using bivariate beta distribution as baseline distribution. This method has opened horizons for development of new bivariate distributions. In this article a bivariate transmuted family of distributions is proposed and some of its properties are studied. The family of distributions is proposed in the following sections.
The Bivariate Transmuted Family of Distributions
Alzaatreh et al. [2] have proposed a new method to derive families of distributions using two different distributions. The proposed family is referred to as the T–X family of distributions. The cdf of this family is
The probability density function (pdf) corresponding to (1) is
fT−X(x)=[ddxW{G(x)}]r[W{G(x)}].
The transmuted family of distributions, proposed by Shaw et al. [1], can be obtained from (1) by using a suitable r(t) with support on [0,1] and W[G(x)]=G(x). Alizadeh et al. [11] has shown that the transmuted family of distribution can be obtained by using r(t)=1+λ−2λt and W[G(x)]=G(x) in (1).
In this article our focus is to extend the transmuted family of distributions to bivariate case. It is, therefore, suitable to discuss the bivariate T–X family of distributions and then use it to propose a new family of distribution called the bivariate transmuted family of distributions.
A simple bivariate extension of (T−X) family of distributions is defined by the joint cdf.
FX,Y(x,y)=∫a1W1[G1(x)]∫a2W2[G2(y)]r(u1,u2)du1du2,
where W1[G1(x)] and W2[G2(y)] have usual properties and r(u1,u2) is any bivariate distribution with suitable support for random variables U1 and U2. If r(u1,u2) is a bivariate distribution such that the support of u_{1} and u_{2} is [0,1]×[0,1] then a simpler version of bivariate T – X family is given as
FX,Y(x,y)=∫0G1(x)∫0G2(y)r(u1,u2)du1du2.
It can be seen that the Gumbel bivariate family of distributions can be obtained from (2) by using
r(u1,u2)=1+γ(1−2u1)(1−2u2);0≤u1,u2≤1.
The bivariate transmuted family of distributions is obtained by using
We will, now, give some properties of the bivariate transmuted family of distributions.
Properties of Bivariate Transmuted Family of Distributions
In this section some important properties of the bivariate transmuted family of distributions are studied. These properties include the marginal and conditional distributions, conditional moments, product and ratio moments, bivariate reliability and bivariate hazard rate functions. Maximum likelihood estimation of the parameters of the family is also given when all the parameters of the baseline distribution are known.
The Marginal and Conditional Distributions
The marginal cdf’s of X and Y are readily written from (3) as
FX(x)=G1(x)[1+(λ1+λ3){1−G1(x)}]
and
FY(y)=G2(y)[1+(λ2+λ3){1−G2(y)}],
where G1(x) and G2(y) are the cdf’s of baseline distributions and (λ1,λ2,λ3)∈[−1,1], −1≤λ1+λ3≤1 and −1≤λ2+λ3≤1. It is easy to see that (5) and (6) are cdf’s of transmuted family of distributions. The marginal density functions of X and Y are
fX(x)=g1(x)[1+(λ1+λ3)−2(λ1+λ3)G1(x)]
and
fY(y)=g2(y)[1+(λ2+λ3)−2(λ2+λ3)G2(y)],
where g1(x) and g2(y) are the pdf’s of baseline distributions corresponding to G1(x) and G2(y), respectively.
The conditional distribution of X given Y=y and Y given X=x for the bivariate transmuted family of distributions are readily written as
where μy−s is sth negative moment of Y and μy(2:2)−s is sth negative moment of larger observation in a sample of size 2 from G2(y).
The Bivariate Reliability and Hazard Rate Functions
The reliability function indicates the probability that a patient, device or other element of interest has ability to survive after some specific time (for more details see [12,13]). The bivariate reliability function for random variables X and Y is defined as
R(x,y)=1−[FX(x)+FY(y)−FX,Y(x,y)].
Now using (5), (6) and (3) in above equation, the bivariate reliability function for the bivariate transmuted family of distributions is
which can be obtained for different baseline cdf’s G1(x) and G2(y).
The hazard rate function is important in reliability studies. It describes the instantaneous rate of failure at any given time. The bivariate hazard rate function (see for example [14]) is defined as
hX,Y(x,y)=fX,Y(x,y)RX,Y(x,y).
The bivariate hazard rate function for the bivariate transmuted family of distributions is obtained by using (4) and (15) in above equation and is
The bivariate hazard rate function can be computed for G1(x) and G2(y).
Dependence
The dependence between two random variables is an important measure to study the relationship between two variables. In this section two improtant dependence measures for the bivariate transmuted family of distributions are obtained. These dependence measures include Kendall’s tau and Spearman’s rho. The Kendall’s tau coefficient for two continuous random variables is computed by using
τ=4∫−∞∞∫−∞∞FX,Y(x,y)fX,Y(x,y)dxdy−1,
which for the bivariate transmuted family of distributions is
τ=−29(λ1+λ3)(λ2+λ3).
Again, the Spearman’s rho for two continuous random variables is obtained as
ρ=12∫−∞∞∫−∞∞[FX,Y(x,y)−FX(x)FY(y)]fX(x)fY(y)dxdy,
which for the bivariate transmuted family of distributions is given as
ρ=−13(λ1+λ3)(λ2+λ3).
We can readily see that the Kendall’s tau will always be larger than the Spearman’s rho for the bivariate transmuted family of distributions.
Estimation of the Parameters
In this section the maximum likelihood estimation of the parameters is done for the bivariate transmuted family of distributions under the assumption that all the parameters of baseline distributions G1(x) and G2(y) are known. For this, suppose X1,X2,⋯,Xn is a random sample of size n from the bivariate transmuted family of distributions. The likelihood function is
The maximum likelihood estimators of λ1, λ2 and λ3 are obtained by equating (20), (21) and (22) to zero and numerically solving the resulting equations.
The bivariate transmuted family of distributions can be explored for any baseline distribution. In the following section we will study a member of the bivariate transmuted family of distributions for baseline Weibull distribution. The resulting distribtuion is named as bivariate transmuted Weibull distribution.
The Bivariate Transmuted Weibull Distribution
The Weibull distribution is a widely used distribution in statistical analysis, especially in the field of reliability. The Weibull distribution, introduced by Weibull [15], has been explored by several authors. Various extensions of the Weibull distribution have been proposed from time-to-time. The transmuted Weibull (TW for short) distribution has proposed by Aryal and Tsokos [16].
In the following we will propose the bivariate transmuted Weibull distribution by using cdf of Weibull distribution in the bivariate transmuted family of distributions. Distributional properties of the proposed distribution will be studied. Parameter estimation of the proposed distribution will also be discussed alongside real data application.
The bivariate transmuted Weibull distribution is obtained by using following cdf of Weibull distribution for X and Y
where 0$]]>(θ1,θ2)>0 are scale parameters, 0$]]>(α1,α2)>0 are shape parameters and (λ1,λ2,λ3) are the transmutation parameters such that all λ’sλ′s∈[−1,1].
Properties of Bivariate Transmuted Weibull Distribution
In this section some statistical properties of the BTW distribution are studied.
The Marginal and Conditional Distributions
The marginal cdf’s of X and Y are immediately written from (24) as
FTW(x)=[1−e−(x/xθθ1)α1][1+(λ1+λ3)e−(x/xθθ1)α1],
and
FTW(y)=[1−e−(y/yθθ2)α2][1+(λ2+λ3)e−(y/yθθ2)α2].
It can be easily seen that both of the marginal cdf’s are transmuted Weibull distributions. The marginal pdf’s of X and Y are obtained from (25) as
The conditional pdf of X given Y=y for the BTW distribution is obtained by using the density and distribution function of Weibull random variable in (9) and is
Similarly, the conditional pdf of Y given X=x for the BTW distribution is obtained by using density and distribution function of Weibull distribution in (10) and is
where b=λ1+λ3 and c=λ2+λ3. The conditional distributions are useful in obtaining the conditional moments of the distribution which will be obtained in the following subsection.
The Conditional Moments
We have seen in (11) and (12) that the conditional moments for the bivariate transmuted family of distributions involve raw moments and moments of order statistics for the baseline distribution. We know that the rth raw moment and rth moment of larger observation in a sample of size 2 from Weibull distribution are, respectively, given as
E(Wr)=θrΓ(1+rα)
and
E(W2:2r)=2θrΓ(1+rα)(1−2−(r/rαα+1)).
Now using (32) and (33) in (11), the rth conditional moment of X given Y=y for the BTW distribution is
where Δ1(y)=1+c−2c{1−e−(y/yθθ2)α2} and Δ2(x)=1+b−2b{1−e−(x/xθθ1)α1}. The conditional moments are useful to obtain the conditional mean and conditional variance of the distribution.
The Product and Ratio Moments
The (r,s)th product moment for the BTW distribution are obtained by using simple moments and moments of maximum in a sample of size 2 from the Weibull distribution in (13). Now, using these moments, the expression for (r,s)th product moment for the BTW distribution is
Again using the raw moments and moments of order statistics from Weibull distribution in (14), the (r,s)th ratio moment for the BTW distribution is obtained as
The Bivariate Reliability and Hazard Rate Functions
The bivariate reliability function of X and Y can be obtained by using (15). The expression for bivariate reliability function for the BTW distribution is
Now using the bivariate density function (25) and bivariate reliability function (38) in (16), the bivariate hazard rate function for the BTW distribution is obtained as
where b and c are defined earlier. The plots of bivariate hazard rate function for θ1=1,θ2=1,λ1=0.25,λ=−0.25 and λ3=0.5 at varios combinations of α1 and α2 are given in Fig. 1 below.
Hazard rate function of bivariate transmuted Weibull distribution
We can see that the distribution has decreasing hazard rate for (α1,α2)<1 and has increasing hazard rate for 1$]]>(α1,α2)>1. The bivariate reliability and bivariate hazard rate functions can be computed and plotted for different other combinations of the parameters.
Parameter Estimation for Bivariate Transmuted Weibull Distribution
In this section the maximum likelihood estimation of the parameters of BTW distribution is given. For this, let X1,X2,…,Xn be a random sample of size n from the BTW distribution. The likelihood function is
The MLEs of the parameters in Θ=(θ1,θ2,α1,α2,λ1,λ2,λ3) are obtained by maximizing the log-likelihood function (40). The derivatives with respect to unknown parameters are
The maximum likelihood estimators of components of parameter vector Θ=(θ1,θ2,α1,α2,λ1,λ2,λ3) are obtained by equating above derivatives to zero and numerically solving the resulting equations.
Real Data Application
In this section two real data applications of the bivariate transmuted Weibull distribution are given. We have modeled two data sets by using the BTW distribution alongside three other bivariate distributions. The distributions that we have used, for comparison, are bivariate Pseudo exponential propose by Mohsin et al. [17], bivariate Weibull propose by Hanif Shahbaz et al. [18] and Gumbel bivariate Weibull distribution. The distributions are fitted by obtaining the maximum likelihood estimates of the parameters. The maximum likelihood estimates of the model parameters are obtained by using the R-package “maxLik” [19]. In order to assess the performance of the BTW with the competing distributions the Akaike’s information criterion (AIC) and Bayesian information criterion (BIC) are computed.
The first data set is related to abortion index. The random variable X in the data set represents the Time and random variable Y represents the Absorption Index. Tab. 1 contains the summary statistics for the data set.
Summary statistics for optical data
Min
Q_{1}
Median
Mean
Q_{3}
Max
X 1.4189
1.7275
2.2573
3.0286
3.4346
9.7145
Y 4.9000e-12
4.6357e-04
7.4168e-04
0.5202
1.6549e-03
1.2344e+01
The results of maximum likelihood estimates are given in Tab. 2. The computed values of AIC and BIC are given in Tab. 3. From Tab. 3, we can see that the bivariate transmuted Weibull distribution has smallest values of AIC and BIC and hence is considered as the best fit for this data.
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</alternatives></inline-formula> for the selected models
Distribution
Parameter
Estimate
SE
θ_{1}
0.0032
0.0019
θ_{2}
0.0042
0.0008
α_{1}
0.4014
0.0309
α_{2}
0.3766
0.0217
Bivariate Transmuted Weibull
λ_{1}
0.5107
0.0205
λ_{2}
0.1387
0.0943
λ_{3}
0.8438
0.0323
Bivariate Pseudo Exponential
α
0.3302
0.0660
Bivariate Weibull
β_{1}
0.7204
0.0776
β_{2}
0.2237
0.0326
α_{1}
1.7595
0.2631
α_{2}
0.2617
0.0372
Gumbel Bivariate Weibull
β_{1}
9.5022
4.6777
β_{2}
0.2855
0.0694
b
0.9999
1.7169
Selection criteria for the selected models
Distribution
LogLik
AIC
BIC
Bivariate Transmuted Weibull
647.0981
–1280.1962
–1271.6641
Bivariate Pseudo Exponential
–48.7663
99.5326
100.7515
Bivariate Weibull
32.56451
–61.1290
–58.6913
Gumbel Bivariate Weibull
61.2213
–112.4425
–106.3482
Atom Arrays Data
In the second data set X represents the Light Emissions and Y represents the Dipole Blockade of atom arrays. Tab. 4 gives some descriptive statistics of the data. The results of maximum likelihood estimates are given in Tab. 5 and computed values of AIC and BIC are listed in Tab. 6. According to Tab. 6, it is obvious that the bivariate transmuted Weibull distribution is the best fit for the second data as it has the smallest values of AIC and BIC.
Summary statistics for atom array data
Min.
Q_{1}
Median
Mean
Q_{3}
Max.
X 0.0095
0.0132
0.0152
0.1296
0.0415
1.3413
Y 0.0094
0.0131
0.0153
0.1309
0.0454
1.3559
<italic>MLEs</italic> and <italic>SEs</italic> for the selected models
Distribution
Parameter
Estimate
SE
θ_{1}
1.2451
0.0158
θ_{2}
3.1611
0.0635
α_{1}
0.9730
0.0652
α_{2}
2.5970
0.1162
Bivariate Transmuted Weibull
λ_{1}
0.7003
0.0887
λ_{2}
–0.0709
0.0031
λ_{3}
–0.7237
0.0687
Bivariate Pseudo Exponential
α
7.7150
1.6450
Bivariate Weibull
β_{1}
0.2004
0.0374
β_{2}
0.3272
0.0645
α_{1}
0.5996
0.0897
α_{2}
0.5980
0.0895
Gumbel Bivariate Weibull
β_{1}
0.2144
0.0572
β_{2}
0.2164
0.0576
b
0.9999
1.4543
Selection criteria for the selected models
Distribution
LogLik
AIC
BIC
Bivariate Transmuted Weibull
325.0031
–636.0062
–628.3689
Bivariate Pseudo Exponential
–56.5817
115.1634
116.2545
Bivariate Weibull
21.5436
–39.0873
–36.9052
Gumbel Bivariate Weibull
69.5337
–129.0675
–123.6122
Conclusions
In this paper we have introduced a new bivariate family of transmuted distributions. The proposed family has been studied for Weibull baseline distribution giving rise to bivariate transmuted Weibull distribution. Various properties of the proposed family and bivariate transmuted Weibull distribution have been studied. We have also applied the bivariate transmuted Weibull distribution on two real data sets. We have seen that the proposed bivariate Weibull distribution turned out to be the best fit for modeling of the data used. The proposed bivariate transmuted family of distributions can be further explored for different other baseline distributions which can be useful in modeling of complex bivariate data.
Funding Statement: The author(s) received no specific funding for this study.
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.
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