Bivariate Beta–Inverse Weibull Distribution: Theory and Applications

Probability distributions have been in use for modeling of random phenomenon in various areas of life. Generalization of probability distributions has been the area of interest of several authors in the recent years. Several situations arise where joint modeling of two random phenomenon is required. In such cases the bivariate distributions are needed. Development of the bivariate distributions necessitates certain conditions, in a field where few work has been performed. This paper deals with a bivariate beta-inverse Weibull distribution. The marginal and conditional distributions from the proposed distribution have been obtained. Expansions for the joint and conditional density functions for the proposed distribution have been obtained. The properties, including product, marginal and conditional moments, joint moment generating function and joint hazard rate function of the proposed bivariate distribution have been studied. Numerical study for the dependence function has been implemented to see the effect of various parameters on the dependence of variables. Estimation of the parameters of the proposed bivariate distribution has been done by using the maximum likelihood method of estimation. Simulation and real data application of the distribution are presented.


Introduction
The probability distributions are widely used in many areas of life. Standard probability distributions have been extended by various authors to increase the applicability of a given baseline distribution. The exponentiated distributions, proposed by Gupta et al. [1], extend a baseline distribution by exponentiation. The beta generated distributions, proposed by Eugene et al. [2], generalize a baseline distribution by using logit of the beta distribution. The cumulative distribution function (cdf) of the Beta-G distributions is where G x ð Þ is the cdf of any baseline distribution and B a; b ð Þ is the complete beta function defined as B a; b ð Þ ¼ The density function of any member of the Beta-G family of distributions is written as Another method of generating new distributions has been proposed by Cordeiro et al. [3] by using the cdf of Kumaraswamy distribution, proposed by Kumaraswamy et al. [4]. The proposed family is referred to as the Kumaraswamy-G (Kum-G) family of distributions and its cdf is A general method of generating new distributions has been proposed by Alzaatreh et al. [5] by using the cdf of any distribution and this method is known as the T-X family of distributions. The cdf of new distribution by using the T-X family of distributions is where r t ð Þ is the density of any random variable defined on d 1 ; d 2 ½ , where d 1 can be -∞ and d 2 can be +∞, and W G x ð Þ ½ is any function of G x ð Þ such that W 0 ð Þ ¼ d 1 and W 1 ð Þ ¼ d 2 .
The Beta-G, the Kum-G and the T-X family of distributions provide basis to obtain a new univariate distribution by using the cdf, G x ð Þ, of any baseline distribution. Different authors have proposed several new distributions by using these families, for example beta-normal by Eugene et al. [2], beta-Weibull by Famoye et al. [6], beta-inverse Weibull by Hanook et al. [7], Kum-inverse Weibull by Shahbaz et al. [8], gammanormal by Alzaatreh et al. [9], exponentiated-gamma distribution by Nadarajah et al. [10], among others.
Several situations arise where we are interested in the simultaneous modeling of two random variables, for example we may be interested in the simultaneous study of arrival and departure time of the customers at a service station. In such situations some suitable bivariate distribution is required. The bivariate beta distribution, proposed by Olkin et al. [11], is a useful bivariate distribution to model random variables which represent proportion of some events of interest. The density function of the bivariate beta distribution is where B a; b; c ð Þis extended beta function defined as and À a ð Þ is the complete gamma function defined as À a ð Þ ¼ The bivariate beta distribution has been used by Sarabia et al. [12] to propose the bivariate Beta-G 1 G 2 (BB-G 1 G 2 ) family of distributions. The joint cdf of the proposed family is for a 1 ; a 2 ; b ð Þ> 0. The density function of the BB-G 1 G 2 family of distributions is and a 1 ; a 2 ; b ð Þ> 0. It is also shown by Sarabia et al. [12] that the density (8) can be written as where f BetaÀG x; a; b ð Þis the density function of the beta-G distribution, given in (2).
The BB-G 1 G 2 family of distributions has not be explored and in this paper we will propose a new bivariate beta-inverse Weibull distribution. The structure of the paper is given below.
A new bivariate beta-inverse Weibull distribution is proposed in Section 2 alongside its marginal and conditional distributions. Some properties of the proposed distribution are obtained in Section 3. Estimation of the parameters of the proposed bivariate distribution is given in Section 4. Section 5 contains simulation and real data applications. Conclusions and recommendations are given in Section 6.

The Bivariate Beta-Inverse Weibull Distribution
The inverse Weibull distribution, also known as the Fréchet [13] distribution, is a useful lifetime distribution. The density and distribution functions of the inverse Weibull distribution are and G x ð Þ ¼ exp Àbx Àa ð Þ; x; b; a > 0; (11) where b is the rate and a is the shape parameter. A random variable X having the inverse Weibull distribution is written as IW a; b ð Þ. Now, suppose that the random variable X has IW Using the distribution functions of X and Y in (7), the distribution function of the bivariate beta-inverse Weibull (BBIW for short) distribution is where B x;y a 1 ; is the extended incomplete beta function and is the extended incomplete beta function ratio. The density function of the BBIW distribution is written from (8) as and b 1 ; a 1 ; b 2 ; a 2 ð Þ> 0. The random variables X and Y having joint density (15) is written as The plots of the density function for b 1 ¼ b 2 ¼ 1 and for different choices of the other parameters are shown in Fig. 1 below.
The joint hazard rate function of the distribution is obtained by using and is given as The hazard rate function can be computed for different values of the parameters.
The density function of the BBIW distribution can also be written in the form of beta-inverse Weibull density functions as where f BetaÀIW x; h ð Þ is the density function of the beta-inverse Weibull random variable with parameter vector h.
The marginal distributions of X and Y are obtained, from (15), below.
Making the transformation exp Àb 2 y Àa 2 ð Þ¼v, we have The conditional distribution of Y given X is readily obtained from (15) and (17) and is Similarly, the conditional distribution of X given Y is The conditional distributions are useful in computing conditional moments.
In the following we will obtain some useful expansions for the joint density function of BBIW distribution. For this, we will use following expansions, for any real a, Now, the density function of the BBIW distribution is Using following series expansion Re-arranging, the above density can be written as where f BetaÀIW x; a; b ð Þis the density function of beta-inverse Weibull distribution. Also , it is easy to see that the joint density function of the BBIW distribution is the weighted sum of product of marginal density functions of the beta-inverse Weibull distributions.
The expansion of the density function, given in (20), can also be written as the weighted sum of density functions of the inverse Weibull distributions. For this, we use following expansions Using these expansions in (20), the density function of the BBIW distribution is written as or Þis the density function of the inverse Weibull random variable with parameters b; a ð Þ and From (21), we can see that the density function of the BBIW distribution is the weighted sum of product of the inverse Weibull density functions. The expression (21) is useful in computing moments of the distribution.
The expansion for the conditional density function of X given Y is readily obtained by using where The expansion of the conditional density function of Y given X is similarly written as The expansions (22) and (23) are helpful in computing the conditional moments of the distribution.

Properties of the Distribution
In this section some properties of the BBIW distribution are discussed. These properties include joint, marginal and conditional moments and are discussed in the following sub-sections.

Product and Ratio Moments of the Distribution
The product and ratio moments of the BBIW distribution are obtained by using the joint density function, given in (15), or the expansion of the density function, given in (21). It is easier to obtain the product and ratio moments from (21) and are obtained below.
The (r,s)th product moment for two random variables is defined as x r y s f X ;Y x; y ð Þdxdy: Using the joint density function of the BBIW distribution, the expression for the product moment is the expression for the product moment for the BBIW distribution is which exists for r < a 1 and s < a 2 .
The (r,s)th ratio moment for two random variables is defined as x r y Às f X ;Y x; y ð Þdxdy: Using the joint density function of the BBIW distribution, the ratio moment is the expression of the ratio moment for the BBIW distribution is Similarly, the expression for the ratio moment, l = Àr;s , for the BBIW distribution is It is to be noted that the parameters a 1 , a 2 and b has no effect on the means and variances of the random variables X and Y as these are controlled by parameters a 1 ; a 2 ; b 1 and b 2 .

Conditional Moments of the Distribution
The conditional moment of a distribution is useful in studying the behavior of one variable under the conditions on the other variable. In case of a bivariate distribution we can compute conditional moment of X given Y = y and of Y given X = x. In the following, the conditional moments for the BBIW distribution are obtained.
The conditional moment of X given Y = y is computed as Now, using the conditional distribution of X given Y = y, from (22), we have Using the density function of the beta-inverse Weibull distribution, we have Now, using the expansion where Similarly, the expression for the conditional moment of Y given X = x is where The conditional means and variances can be obtained from (27) and (28).

The Dependence Function and Correlation Coefficient
The dependence function for a bivariate distribution is, given by Holland et al. [14], and Sarabia et al. [12] has shown that the dependence function for the BB-G 1 G 2 family is Using the density and distribution functions of the inverse Weibull distribution in (29), the dependence function for the BBIW distribution is which is always positive. It is to be noted that the dependence function is different from the linear correlation coefficient which is to be computed from the product moment. The values of correlation coefficient for different Table 1: Correlation coefficient between X and Y for the BBIW distribution combinations of the parameters are given in Tab. 1 below. The values of correlation coefficient indicate that the effect of a 1 and a 2 on the correlation coefficient is positive, that is increase in the values of a 1 and a 2 will increase the linear correlation between X and Y for the BBIW distribution. The effect of other parameters on the correlation coefficient is negative, that is increase in b, a 1 and a 2 will decrease the correlation coefficient.

The Shannon Entropy
The Shannon entropy, Shannon [15], in a bivariate distribution is defined as Now, for the BBIW distribution we have ð Þln 1 À exp Àb 1 x Àa 1 ð Þexp Àb 2 y Àa 2 ð Þ ½ :  ð Þ ½ n a n 1 a n 2 b n 1 b n 2 x a 1 þ1 y a 2 þ1 exp Àa 1 b 1 The derivatives of the log-likelihood function with respect to the unknown parameters are Results of the fitted models are given in Tab. 3 below.  From the table, we can see that the BBIW distribution is the best fit to the data as it has smallest AIC. The fitted BBIW distribution is shown in Fig. 4 below.
The graph of the fitted BBIW distribution is reasonably close to the bivariate histogram of the data.

Conclusions
In this paper a new bivariate beta inverse Weibull distribution is proposed by using the logit of bivariate beta distribution. The distribution is very flexible and is useful in modeling of the complex data. The properties of the distribution have been studied and it is found that the correlation coefficient is controlled by combination of three parameters a 1 , a 2 and b. The proposed bivariate beta inverse Weibull distribution has been fitted to the world GNI data of two years and it is found that the proposed BBIW is better fit as compared with the other models involved in the study

Conflicts of Interest:
The authors declare no conflict of interest regarding the publication of this paper.