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

(1) FBeta−G(x)=1B(a,b)∫0G(x)ua−1(1−u)b−1du,0<G(x)<1,(a,b)>0,

where G(x) is the cdf of any baseline distribution and B(a,b) is the complete beta function defined as

B(a,b)=∫01ua−1(1−u)b−1du.

The density function of any member of the Beta–G family of distributions is written as

(2) fBeta−G(x;a,b)=1B(a,b)g(x)[G(x)]a−1[1−G(x)]b−1,0<G(x)<1.

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

(3) FKum−G(x)=1−[1−Ga(x)]b,0<G(x)<1,a,b>0.

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

(4) FT−X(x)=∫d1W[G(x)]r(t)dt,

where r(t) is the density of any random variable defined on [d1,d2] , where d₁ can be –∞ and d₂ can be +∞, and W[G(x)] is any function of G(x) such that W(0)=d1 and W(1)=d2 .

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) , 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], gamma-normal 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

(5) fX,Y(x,y)=xa−1yb−1(1−x)b+c−1(1−y)a+c−1B(a,b,c)(1−xy)a+b+c,0<(x,y)<1,

where B(a,b,c) is extended beta function defined as

(6) B(a,b,c)=∫01∫01ua−1vb−1(1−u)b+c−1(1−v)a+c−1(1−uv)a+b+cdudv,

and Γ(a) is the complete gamma function defined as

Γ(a)=∫0∞ua−1e−udu.

The bivariate beta distribution has been used by Sarabia et al. [12] to propose the bivariate Beta-G₁G₂ (BB-G₁G₂) family of distributions. The joint cdf of the proposed family is

(7) FBeta−G1(x)G2(y)(x,y)=FBeta−G1G2(x,y)=1B(a1,a2,b)∫0G2(y)∫0G1(x)ua1−1va2−1(1−u)a2+b−1(1−v)a1+b−1(1−uv)a1+a2+bdudv,

for 0$]]>(a1,a2,b)>0 . The density function of the BB-G₁G₂ family of distributions is

(8) fBeta−G1G2(x,y)=1B(a1,a2,b)g1(x)g2(y)[G1(x)]a1−1[G2(y)]a2−1[1−G1(x)]a2+b−1×[1−G2(y)]a1+b−1[1−G1(x)G2(y)]−(a1+a2+b),0<G1(x),G2(y)<1,

and 0$]]>(a1,a2,b)>0 . It is also shown by Sarabia et al. [12] that the density (8) can be written as

(9) fBeta−G1G2(x,y)=Γ(a1+b)Γ(a2+b)Γ(b)Γ(a1+a2+b)fBeta−G1(x;a1,a2+b)fBeta−G2(y;a2,a1+b)[1−G1(x)G2(y)](a1+a2+b),

where fBeta−G(x;a,b) is the density function of the beta-G distribution, given in (2).

The BB-G₁G₂ 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 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

(10) 0,$$]]>g(x)=βαxα+1exp⁡(−βx−α),x,β,α>0,

and

(11) 0,$$]]>G(x)=exp⁡(−βx−α),x,β,α>0,

where β is the rate and α is the shape parameter. A random variable X having the inverse Weibull distribution is written as IW(α,β) . Now, suppose that the random variable X has IW(α1,β1) and the random variable Y has IW(α2,β2) distribution with cdf’s

G1(x)=exp⁡(−β1x−α1)andG2(y)=exp⁡(−β2y−α2).

Using the distribution functions of X and Y in (7), the distribution function of the bivariate beta-inverse Weibull (BBIW for short) distribution is

FX,Y(x,y)=1B(a1,a2,b)∫0exp⁡(−β2y−α2)∫0exp⁡(−β1y−α1)ua1−1va2−1(1−u)a2+b−1(1−v)a1+b−1(1−uv)a1+a2+bdudv,

or

(12) FX,Y(x,y)=Iexp⁡(−β1xα1),exp⁡(−β2yα2)(x,y)=Bexp⁡(−β1xα1),exp⁡(−β2yα2)(a1,a2,b)B(a1,a2,b),

where

(13) Bx,y(a1,a2,b)=∫0x∫0yua1−1va2−1(1−u)a2+b−1(1−v)a1+b−1(1−uv)a1+a2+bdudv

is the extended incomplete beta function and

(14) Ix,y(a1,a2,b)=Bx,y(a1,a2,b)B(a1,a2,b)

is the extended incomplete beta function ratio. The density function of the BBIW distribution is written from (8) as

(15) 0,}$$]]>fX,Y(x,y)=1B(a1,a2,b)α1α2β1β2xα1+1yα2+1exp⁡(−a1β1x−α1)exp⁡(−a2β2y−α2)[1−exp⁡(−β1x−α1)]a2+b−1×[1−exp⁡(−β2y−α2)]a1+b−1×[1−exp⁡(−β1x−α1)exp⁡(−β2y−α2)]−(a1+a2+b),(x,y)>0,

and 0$]]>(β1,α1,β2,α2)>0 . The random variables X and Y having joint density (15) is written as BBIW(β1,β2,α1,α2,a1,a2,b) . The plots of the density function for β1=β2=1 and for different choices of the other parameters are shown in Fig. 1 below.

The plot of joint distribution function for β1=β2=1 , α1=2.5 , α2=1.5 , a1=1.5 , a2=2.5 and b=2.0 is given in Fig. 2.

The BBIW distribution provides various other distributions as special case. For example, for α1=α2=1 we have the bivariate beta inverse exponential (BBIE) distribution and for α1=α2=2 the distribution reduces to the bivariate beta inverse Rayleigh (BBIR) distribution.

The joint hazard rate function of the distribution is obtained by using

λ(x,y)=fX,Y(x,y)1−FX,Y(x,y)

and is given as

(16) λ(x,y)={α1α2β1β2xα1+1yα2+1exp⁡(−a1β1x−α1)exp⁡(−a2β2y−α2)[1−exp⁡(−β1x−α1)]a2+b−1×[1−exp⁡(−β2y−α2)]a1+b−1[1−exp⁡(−β1x−α1)exp⁡(−β2y−α2)]−(a1+a2+b)}÷[B(a1,a2,b)−Bexp⁡(−β1xα1),exp⁡(−β2yα2)(a1,a2,b)].

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

0,$$]]>fX,Y(x,y)=Γ(a1+b)Γ(a2+b)Γ(b)Γ(a1+a2+b)fBeta−IW(x;a1,a2+b,β1,α1)fBeta−IW(y;a2,a1+b,β2,α1)[1−exp⁡(−β1x−α1)exp⁡(−β2y−α2)](a1+a2+b),(x,y)>0,

where fBeta−IW(x;θ) is the density function of the beta-inverse Weibull random variable with parameter vector θ .

The marginal distributions of X and Y are obtained, from (15), below.

Now

fX(x)=1B(a1,a2,b)α1β1xα1+1exp⁡(−a1β1x−α1)[1−exp⁡(−β1x−α1)]a2+b−1∫0∞α2β2yα2+1exp⁡(−a2β2y−α2)×[1−exp⁡(−β2y−α2)]a1+b−1[1−exp⁡(−β1x−α1)exp⁡(−β2y−α2)]−(a1+a2+b)dy

Making the transformation exp⁡(−β2y−α2)=v , we have

fX(x)=1B(a1,a2,b)α1β1xα1+1exp⁡(−a1β1x−α1)[1−exp⁡(−β1x−α1)]a2+b−1∫01va2(1−v)a1+b−1[1−exp⁡(−β1x−α1)v]a1+a2+bdv

or

(17) 0,$$]]>fX(x)=1B(a1,b)α1β1xα1+1exp⁡(−a1β1x−α1)[1−exp⁡(−β1x−α1)]b−1,x>0,

which is density function of BIW(a1,b,β1,α1) . Similarly, it can be shown that the marginal distribution of Y is BIW(a2,b,β2,α2) .

The conditional distribution of Y given X is readily obtained from (15) and (17) and is

(18) 0. \cr} $$]]>fY|X(y|x)=1B(a1,a2,b)α2β2yα2+1exp⁡(−a2β2y−α2)[1−exp⁡(−β2y−α2)]a1+b−1[1−exp⁡(−β1x−α1)]a2×[1−exp⁡(−β1x−α1)exp⁡(−β2y−α2)]−(a1+a2−b),(x,y)>0.

Similarly, the conditional distribution of X given Y is

(19) 0. \cr} $$]]>fX|Y(x|y)=1B(a1,a2,b)α1β1xα1+1exp⁡(−a1β1x−α1)[1−exp⁡(−β1x−α1)]a2+b−1[1−exp⁡(−β2y−α2)]a1×[1−exp⁡(−β1x−α1)exp⁡(−β2y−α2)]−(a1+a2−b),(x,y)>0.

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,

(1−z)−a=∑j=0∞Γ(a+j)j!Γ(a)zjand(1−z)a=∑j=0∞(−1)jΓ(a+1)j!Γ(a−j+1)zj.

Now, the density function of the BBIW distribution is

fX,Y(x,y)=1B(a1,a2,b)α1α2β1β2xα1+1yα2+1exp⁡(−a1β1x−α1)exp⁡(−a2β2y−α2)[1−exp⁡(−β1x−α1)]a2+b−1×[1−exp⁡(−β2y−α2)]a1+b−1[1−exp⁡(−β1x−α1)exp⁡(−β2y−α2)]−(a1+a2+b).

Using following series expansion

[1−exp⁡(−β1x−α1)exp⁡(−β2y−α2)]−(a1+a2+b)=∑j=0∞Γ(a1+a2+b+j)j!Γ(a1+a2+b)exp⁡(−jβ1x−α1)exp⁡(−jβ2y−α2)

we have

fX,Y(x,y)=1B(a1,a2,b)α1α2β1β2xα1+1yα2+1exp⁡(−a1β1x−α1)exp⁡(−a2β2y−α2)[1−exp⁡(−β1x−α1)]a2+b−1×[1−exp⁡(−β2y−α2)]a1+b−1∑j=0∞Γ(a1+a2+b+j)j!Γ(a1+a2+b)exp⁡(−jβ1x−α1)exp⁡(−jβ2y−α2).

Re-arranging, the above density can be written as

fX,Y(x,y)=∑j=0∞dA(j)1B(a1+j,a2+b)α1β1xα1+1exp⁡[−(a1+j)β1x−α1][1−exp⁡(−β1x−α1)]a2+b−1×1B(a2+j,a1+b)α2β2yα2+1exp⁡[−(a2+j)β2y−α2][1−exp⁡(−β2y−α2)]a1+b−1

or

(20) fX,Y(x,y)=∑j=0∞dA(j)fBeta−IW(x;a1+j,a2+b)fBeta−IW(y;a2+j,a1+b),

where fBeta−IW(x;a,b) is the density function of beta-inverse Weibull distribution. Also

d=Γ(a1+b)Γ(a2+b)Γ(a1)Γ(a2)Γ(b)

and

A(j)=Γ(a1+j)Γ(a2+j)Γ(a1+a2+b+j)⋅1j!

From (20), 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

[1−exp⁡(−β1x−α1)]a2+b−1=∑h=0∞(−1)hΓ(a2+b)h!Γ(a2+b−h)exp⁡(−hβ1x−α1)

and

[1−exp⁡(−β2y−α2)]a1+b−1=∑k=0∞(−1)kΓ(a1+b)k!Γ(a1+b−j)exp⁡(−kβ2y−α2).

Using these expansions in (20), the density function of the BBIW distribution is written as

(21) fX,Y(x,y)=∑j=0∞∑h=0∞∑k=0∞dA(j,h,k)α1(a1+j+h)β1xα1+1exp⁡[−(a1+j+h)β1x−α1]×α2(a2+j+k)β2yα2+1exp⁡[−(a2+j+k)β2y−α2],

or

fX,Y(x,y)=∑j=0∞∑h=0∞∑k=0∞dA(j,h,k)fIW(x;(a1+j+h)β1,α1)fIW(y;(a2+j+k)β2,α2),

where fIW(x;β,α) is the density function of the inverse Weibull random variable with parameters (β,α) and

A(j,h,k)=A(j)⋅(−1)h+kh!k!⋅Γ(a1+b)Γ(a1+b−h)⋅Γ(a2+b)Γ(a2+b−k)⋅1(a1+j+h)(a2+j+k).

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

[1−exp⁡(−β1x−α1)exp⁡(−β2y−α2)]−(a1+a2+b)=∑j=0∞Γ(a1+a2+b+j)j!Γ(a1+a2+b)exp⁡(−jβ1x−α1)exp⁡(−jβ2y−α2)

in (19) and is

fX|Y(x|y)=1B(a1,a2,b)α1β1xα1+1exp⁡(−a1β1x−α1)[1−exp⁡(−β1x−α1)]a2+b−1[1−exp⁡(−β2y−α2)]a1×∑j=0∞Γ(a1+a2+b+j)j!Γ(a1+a2+b)exp⁡(−jβ1x−α1)exp⁡(−jβ2y−α2),

or

fX|Y(x|y)=Γ(a2+b)Γ(a1)Γ(a2)Γ(b)∑j=0∞Γ(a1+j)j!1B(a1+j,a2+b)α1β1xα1+1exp⁡[−(a1+j)β1x−α1]×[1−exp⁡(−β1x−α1)]a2+b−1exp⁡(−jβ2y−α2)[1−exp⁡(−β2y−α2)]a1,

or

(22) fX|Y(x|y)=[1−exp⁡(−β2y−α2)]a1∑j=0∞d1A1(j)fBeta−IW(x;a1+j,a2+b)exp⁡(−jβ2y−α2),

whered1=Γ(a2+b)Γ(a1)Γ(a2)Γ(b)andA1(j)=Γ(a1+j)j!.

The expansion of the conditional density function of Y given X is similarly written as

(23) fY|X(y|x)=[1−exp⁡(−β1x−α1)]a2∑j=0∞d2A2(j)fBeta−IW(y;a2+j,a1+b)exp⁡(−jβ1x−α1),

whered2=Γ(a1+b)Γ(a1)Γ(a2)Γ(b)andA2(j)=Γ(a2+j)j!.