DOI:10.32604/csse.2021.014764
| Computer Systems Science & Engineering DOI:10.32604/csse.2021.014764 | |
| Article |
The Bivariate Transmuted Family of Distributions: Theory and Applications
Department of Statistics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia
*Corresponding Author: Muhammad Qaiser Shahbaz. Email: mkmohamad@kau.edu.sa
Received: 14 October 2020; Accepted: 06 November 2020
Abstract: 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.
Keywords: Transmuted Distributions; (T−X) family of distributions; bivariate transmuted family of distributions; Weibull distribution; maximum likelihood estimation
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 
of this family is
where 
is the 
of the baseline distribution and 
is the transmutation parameter such that 
.
Various researches have extended the transmuted family of distributions by using the 
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 
of Gumbel bivariate distribution is given as
where 
and 
are any marginal cdf’s and 
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 
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.
2 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 
of this family is
where 
is some function of 
such that
The probability density function (
) corresponding to (1) is
The transmuted family of distributions, proposed by Shaw et al. [1], can be obtained from (1) by using a suitable 
with support on 
and 
. Alizadeh et al. [11] has shown that the transmuted family of distribution can be obtained by using 
and 
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 
family of distributions is defined by the joint cdf.
where 
and 
have usual properties and 
is any bivariate distribution with suitable support for random variables 
and 
. If 
is a bivariate distribution such that the support of u1 and u2 is 
then a simpler version of bivariate T – X family is given as
It can be seen that the Gumbel bivariate family of distributions can be obtained from (2) by using
The bivariate transmuted family of distributions is obtained by using
in (2) and the joint cdf of the proposed bivariate transmuted family is
where 
and 
are any marginal cdf’s and 
are the transmutation parameters such that 
, 
and 
.
The density function corresponding to (3) is
We will, now, give some properties of the bivariate transmuted family of distributions.
3 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.
3.1 The Marginal and Conditional Distributions
The marginal cdf’s of X and Y are readily written from (3) as
and
where 
and 
are the cdf’s of baseline distributions and 
, 
and 
. 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
and
where 
and 
are the pdf’s of baseline distributions corresponding to 
and 
, respectively.
The conditional distribution of 
given 
and 
given 
for the bivariate transmuted family of distributions are readily written as
and
where 
and 
The conditional distributions can be studied for any baseline distribution.
The rth conditional moment of 
given 
is obtained by using
Now using the conditional distribution of 
given 
, given in (9), the rth conditional moment of 
given 
is
where 
, 
is rth raw moment of 
and 
is rth raw moment of larger observation in a sample of size 2 from 
.
Again the sth conditional moment of 
given 
is defined as
Using (10), the sth conditional moment of 
given 
is
where 
is sth raw moment of 
and 
is sth raw moment of larger observation in a sample of size 2 from 
.
3.3 The Product and Ratio Moments
The product and ratio moments are defined as
and
The product moments for the bivariate transmuted family of distribution are obtained as
which on simplification becomes
Similarly, the 
th ratio moments for the bivariate transmuted family of distributions are obtained as
which on simplification becomes
where 
is sth negative moment of 
and 
is sth negative moment of larger observation in a sample of size 2 from 
.
3.4 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
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 
and 
.
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
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 
and 
.
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
which for the bivariate transmuted family of distributions is
Again, the Spearman’s rho for two continuous random variables is obtained as
which for the bivariate transmuted family of distributions is given as
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.
3.6 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 
and 
are known. For this, suppose 
is a random sample of size 
from the bivariate transmuted family of distributions. The likelihood function is
and the log-likelihood function is
The derivatives of log-likelihood function with respect to 
, 
and 
are
and
The maximum likelihood estimators of 
, 
and 
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.
4 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 (
for short) distribution has proposed by Aryal and Tsokos [16].
In the following we will propose the bivariate transmuted Weibull distribution by using 
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 
of Weibull distribution for X and Y
in (3). The 
of bivariate transmuted Weibull (BTW for short) distribution is
The density function of the distribution is
where 
are scale parameters, 
are shape parameters and 
are the transmutation parameters such that all λ’s
.
5 Properties of Bivariate Transmuted Weibull Distribution
In this section some statistical properties of the BTW distribution are studied.
5.1 The Marginal and Conditional Distributions
The marginal cdf’s of 
and 
are immediately written from (24) as
and
It can be easily seen that both of the marginal cdf’s are transmuted Weibull distributions. The marginal pdf’s of 
and 
are obtained from (25) as
and
The conditional 
of 
given 
for the BTW distribution is obtained by using the density and distribution function of Weibull random variable in (9) and is
Similarly, the conditional 
of 
given 
for the BTW distribution is obtained by using density and distribution function of Weibull distribution in (10) and is
where 
and 
. The conditional distributions are useful in obtaining the conditional moments of the distribution which will be obtained in the following subsection.
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
and
Now using (32) and (33) in (11), the rth conditional moment of 
given 
for the BTW distribution is
Similarly, the sth conditional moment of 
given 
is
where 
and 
. The conditional moments are useful to obtain the conditional mean and conditional variance of the distribution.
5.3 The Product and Ratio Moments
The 
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 
th product moment for the BTW distribution is
Again using the raw moments and moments of order statistics from Weibull distribution in (14), the 
th ratio moment for the BTW distribution is obtained as
where 
, b and c are defined earlier.
5.4 The Bivariate Reliability and Hazard Rate Functions
The bivariate reliability function of 
and 
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 
and 
at varios combinations of 
and 
are given in Fig. 1 below.
Figure 1: Hazard rate function of bivariate transmuted Weibull distribution
We can see that the distribution has decreasing hazard rate for 
and has increasing hazard rate for 
. The bivariate reliability and bivariate hazard rate functions can be computed and plotted for different other combinations of the parameters.
6 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 
be a random sample of size 
from the BTW distribution. The likelihood function is
The log-likelihood function is
The 
of the parameters in 
are obtained by maximizing the log-likelihood function (40). The derivatives with respect to unknown parameters are
and
The maximum likelihood estimators of components of parameter vector 
are obtained by equating above derivatives to zero and numerically solving the resulting equations.
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.
7.1 Optical Data
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.
Table 1: Summary statistics for optical data
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.
Table 2: 
and 
for the selected models
Table 3: Selection criteria for the selected models
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.
Table 4: Summary statistics for atom array data
Table 5: MLEs and SEs for the selected models
Table 6: Selection criteria for the selected models
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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