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Volume 18, Issue 3, September 2019, Pages 198 - 211

Bivariate and Multivariate Weighted Kumaraswamy Distributions: Theory and Applications

Authors
Indranil Ghosh*
Department of Mathematics and Statistics, University of North Carolina, Wilmington, 601 S College Road, Wilmington, North Carolina 28403, USA
Corresponding Author
Indranil Ghosh
Received 15 August 2017, Accepted 27 November 2017, Available Online 16 September 2019.
DOI
10.2991/jsta.d.190619.001How to use a DOI?
Keywords
Weighted distributions; Bivariate weighted Kumaraswamy distribution; Renyi entropy; Multivariate weighted Kumaraswamy distribution
Abstract

Weighted distributions (univariate and bivariate) have received widespread attention over the last two decades because of their flexibility for analyzing skewed data. In this paper, we derive the bivariate and multivariate weighted Kumaraswamy distributions via the construction method as discussed in B.C. Arnold, I. Ghosh, A. Alzaatreh, Commun. Stat. Theory Methods. 46 (2017), 8897–8912. Several structural properties of the bivariate weighted distributions including marginals, distributions of the minimum and maximum, reliability parameter, and total positivity of order two are discussed. We provide some multivariate extensions of the proposed bivariate weighted Kumaraswamy model. Two real-life data sets are used to show the applicability of the bivariate weighted Kumaraswamy distributions and is compared with other rival bivariate Kumaraswamy models.

Copyright
© 2019 The Authors. Published by Atlantis Press SARL.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

Recently the construction of continuous bivariate distributions have received a considerable amount of interest in the literature. A vast literature on this topic exists (see, the book by Balakrishnan and Lai [1]). Kumaraswamy [2] argued that the beta distribution does not faithfully fit hydrological random variables such as daily rainfall and daily stream flow and introduced an alternative distribution to the beta distribution, which is known as the Kumaraswamy distribution. According to Nadarajah [3] several papers in the hydrological literature have used this distribution because it is deemed as a better alternative to the beta distribution. The Kumaraswamy distribution is also known as minimax distribution, and generalized beta distribution of the first kind (or beta type I). As a motivation to our current work, we consider a financial risk modeling scenario. In the context of bounded dependent risks, it is desirable to have available flexible models with analytic expressions for the corresponding marginal distributions and densities. In such a context, after rescaling the bounded risks to the interval (0, 1), Kumaraswamy distributions may provide attractive candidate components for such models because of the simplicity of their corresponding density and distribution functions. It thus merits our attention to develop a spectrum of bivariate and multivariate models with Kumaraswamy marginal and/or conditional distributions or at least Kumaraswamy type bivariate and multivariate distributions. For a detailed study, interested readers are referred to Arnold and Ghosh [4], Wagner et al. [5]. The usefulness and applications of weighted distribution to biased samples in various areas including medicine, ecology, reliability, and branching processes can also be seen in Nanda and Jain [6], Gupta and Keating [7], Oluyede [8], Zelen and Feinleib [9], and the references therein. Recently, Arnold et al. [10] introduced a new method of constructing bivariate weighted distributions which they used to model some real-life data sets independently. Al-Mutairi et al. [11] developed a new bivariate distribution with weighted exponential marginals and discussed its multivariate generalization. In this paper, we consider the particular weight function considered Arnold et al. [10], and in addition, namely the maximum conditioning weight function, and via the modified symmetric bivariate Fa–rlieGumbel–Morgenstern (FGM, henceforth, in short) copula. This is completely different from several other methods of obtaining a bivariate Kumaraswamy distribution as discussed in Arnold et al. [4,12,13]. The symmetric bivariate FGM copula based construction approach mentioned in this article, is completely different from Arnold et al. [5], where the authors developed several strategies for constructing bivariate Kumaraswamy type distributions via Arnold–Ng copula. We focus our attention on the application of this new weighted bivariate and multivariate Kumaraswamy distribution. We envision a real-life scenario as a genesis of the proposed bivariate weighted distribution in a stress–strength model context.

Let X,Y be a two-dimensional absolutely continuous random variable with joint density function (with respect to Lebesgue measure) fx,y. Also let Ω,F,P be the common probability space on which X and Y are defined. Using the notation of Arnold and Nagaraja [14], the weighted distribution of X,Y denoted by XW,YW is given by the following joint density function:

f Wx,y=Wx,yfx,yEWX,Y,
where Wx,y is a nonnegative function such that EWX,Y<∞. The utility of such distributions is well established in literature. Rao [15] employed univariate weighted distributions in various real-life problems such as in analysis of family size, aerial survey and visibility bias, renewal theory, cell cycle analysis, efficacy of early screening for disease, statistical ecology, and reliability modeling. An exhaustive amount of work in this area is available in Patil and Rao [16]. Mahfoud and Patil [17] discussed the properties of bivariate weighted densities based on two different choices of the weight function Wx,y, namely xα and maxx,y. Arnold and Nagarajah [14] focused on the case of two independent random variables X,Y and discussed the dependence structure of the corresponding bivariate weighted distributions. In this article, we focused on the maxx,y weight function in more details. In particular, we consider the following different weighted bivariate and multivariate Kumaraswamy models:

  • Assume a system has two independent components with strengths W1 and W2, and suppose that to run the process each component strength has to overcome an outside stress W0 which is independent of both W1 and W2. If we define X,Y=dW1,W2|minW1,W2>W0, where the Wi′s have absolutely continuous distributions, then the resulting joint distribution of W1,W2 is the type of bivariate weighted distribution to be investigated in this article. Henceforth, we call this as Type I-weighted bivariate Kumaraswamy models. The utility of such models have been discussed in details in Arnold et al. [10].

  • The motivation for the second type (from now on, Type-II weighted bivariate Kumaraswamy model) can be described as follows: Consider a two component system, and suppose that, component J, J=1,2 receives outside random shocks, measured as Wi,i=1,2, and they are independent. Next, consider that the minimum of strength among the two components is measured as W0, which is independent of Wi,i=1,2. Furthermore, let us assume that the implicit condition for the associated system to run is that W0 must exceed the maximum of W1 and W2. Then the resulting joint distribution of W1,W2 is the type of bivariate weighted distribution to be investigated next. This is the major motivation to consider this type of models, that we discuss the outside stress structure rather than the inside strength. The genesis of this model is distinct as compared to the previous one, although they might have some similar type properties as we shall see later on.

  • The third method considered is based on a symmetric bivariate modified FGM type copula. Since, the quantile function of a Kumaraswamy distribution is available in simple analytical form, we can use this to construct various bivariate and multivariate Kumaraswamy models. A copula Cx1,x2 is most simply described as a bivariate distribution with Uniform0,1 marginal distributions. A bivariate Kumaraswamy distribution can then be obtained from any copula Cx1,x2, by using marginal transformations of the form indicated earlier. Thus if U1,U2 has the copula Cx1,x2 as its distribution function, then the random vector X1,X2=1−(1−U)1/b11/a1,1−1−U1/b21/a2, will have a bivariate Kumaraswamy distribution in which Xi∼Kai,bi, i=1,2. This approach thus provides us with a plethora of bivariate Kumaraswamy models. However, it is not clear how to sensibly select the particular copula to be used in the construction in achieving greater flexibility. In a separate article, we will focus our attention in this direction. Henceforth, we will call this method as Type-III weighted bivariate and multivariate copula based Kumaraswamy type distribution.

The remainder of this paper is organized as follows: In Section 2, we briefly describe the method of constructing the bivariate weighted distributions. In Section 3, we introduce a special case of the proposed family, the bivariate weighted Kumaraswamy (BWK) distribution and discuss various properties. In Section 4, we discuss a BWK model via conditioning on maximum as mentioned earlier. In Section 5, some discussion on the multivariate extension of the proposed family is provided. Section 6 deals with the estimation of the BWK distribution parameters. For illustrative purposes, two real-life data seta are fitted to the proposed model in Section 7, and are compared with other bivariate Kumaraswamy and bivariate beta models. Some concluding remarks are provided in Section 8.

2. TYPE-I WEIGHTED BIVARIATE KUMARASWAMY DISTRIBUTION

Let W1,W2, and W0 be independent random variables with density functions fWiwi, i=0,1,2. Then according to Arnold et al. [10] if we define X,Y=dW1,W2|minW1,W2>W0 then the density function of the corresponding bivariate weighted distribution is given by

fX,Yx,y=fW1xfW2yPminW1,W2>W0|W1=x,W2=yPminW1,W2>W0 =fW1xfW2yPW0<minx,yPminW1,W2>W0 =fW1xfW2yFW0minx,yPminW1,W2>W0.

Remarks:

  1. If Wi′s, i=0,1,2 are identically distributed with common density function fWw, then PminW1,W2>W0=13. Hence, (1) reduces to

    fX,Y(x,y)=3fW(x)fW(y)FW(min(x,y)).

  2. In general, we do not have a simple expression for PminW1,W2>W0, but in some cases it can be obtained in a closed form. For example, it can be computed if Wi's are independent exponential random variables with intensities λ1,λ2, and λ0.

  3. One can obtain a multivariate extension of (1) in the following way: If Wi∼fWiwi for i=0,1,⋯,k are independent random variables, then the k-dimensional weighted density analogous to (1) will be of the form

    fX1,X2,…Xkx1,x2,⋯,xk=∏j=1kfWjxjFW0minx1,x2,…xkPminW1,W2,…,Wk>W0.

    In the case in which the Wi's are independent and identically distributed (i.i.d.) random variables, (3) reduces to

    fX1,X2,…Xkx1,x2,…xk=k+1∏j=1kfWxjFWminx1,x2,…xk.

In the next section, we study a special case of the bivariate weighted distribution in (1) where the Wi's are Kumaraswamy with parameters a and bi, for i=0,1,2, respectively.

2.1. Definition and Structural Properties

Consider the scenario in which three Wi's are independent random variables with Wi∼Kumaraswamy a,bi for i=0,1,2. Then from (1), the normalizing constant will be

C1=PminW1,W2>W0 =b0b0+b1+b2.

Hence, the joint distribution of X,Y will be

fx,y=C1−1a2b1b2xya−11−xab1−11−yab2−11−1−minx,yab0×I(0<x<1,0<y<1).

In this case the marginals are given by

fx=C1−1ab0b1b0+b2xa−11−xab1−1−1−xab0+b1+b2−1×I(0<x<1).
fy=C1−1ab0b2b0+b1ya−11−yab2−1−1−yab0+b1+b2−1×I(0<y<1).

Let t11, t12, t21, and t22 be real numbers with 0<t11<t12 and 0<t21<t22. Then, X,Y has the total positivity of order two (TP2) property if for any such set of tij's,

fX,Yt11,t21fX,Yt12,t22−fX,Yt12,t21fX,Yt11,t22⩾0.

Theorem 1.

The BWK distribution has the TP2 property.

Proof.

Let us consider different cases separately. If 0<t11<t21<t12<t22, then for the density function in (6), one can easily show that the condition in (9) is equivalent to

1−t21ab0⩾1−t12ab0.

Now, (10) holds because t21<t12 and a and b0 are positive. The other cases can be shown similarly. Hence the proof.

Theorem 2.

The BWK distribution with the density in (6) is log concave.

Proof.

Taking negative of the logarithm of (6), we have the following:

Consider 0<x<y<1

−log fx,y=constant−a−1logx+logy−b1−1log 1−xa  −b2−1log 1−ya−log1−1−xab0.

Next, taking partial double derivative w.r.t x and y, of (11), we get,

DyDx−log fx,y=0,
where Dx stands for the partial derivative operator for x and similarly for Dy.

Case 2: When 0<y<x<1

Similar result will hold here as well. Hence the proof.

The log-concave property implies the following:

  • The Type-I weighted bivariate Kumaraswamy is unimodal.

  • The marginals are also log-concave.

  • It is closed under weak limits.

Next, note that for any r⩾1, the marginal moments of X and Y are given by, respectively

EXr=rΓraΓb1Γra+b1+1−Γb0+b1+b2Γra+b0+b1+b2+1a2,
and
EYr=rΓraΓb2Γra+b2+1−Γb0+b1+b2Γra+b0+b1+b2+1a2.

The correlation coefficient ρ for this distribution is given by

ρ=A1B1B2,
where
A1=EXY−EXEY =2C−11a+1Γ2+2/aΓb2Γ2+2/a+b2−∑k=0∞−1kb0kab0+1+1Γ1+2/a+b0Γb2Γ1+2/a+b2+b0  −C−1(Γ1aΓb1Γ1a+b1+1−Γb0+b1+b2Γ1a+b0+b1+b2+1a2)×ΓraΓb2Γ1a+b2+1−Γb0+b1+b2Γ1a+b0+b1+b2+1a2,
and
B1=VarX =C−12ΓraΓb1Γ2a+b1+1−Γb0+b1+b2Γ2a+b0+b1+b2+1a2−Γ1aΓb1Γ1a+b1+1−Γb0+b1+b2Γ1a+b0+b1+b2+1a22.

Similarly,

B2=VarY =C−12ΓraΓb2Γ2a+b2+1−Γb0+b1+b2Γ2a+b0+b1+b2+1a2−Γ1aΓb2Γ1a+b2+1−Γb0+b1+b2Γ1a+b0+b1+b2+1a22.

Note: From the expression of the correlation coefficient, it is evident that this model will exhibit both positive and negative correlation, depending on the choice of the parameters a,b0,b1,b2. Thus we seek a bivariate weighted distribution with a positive probability on the unit square (0, 1)2, with marginals are of univariate Kumaraswamy type, and correlation over the full range. The bivariate Kumaraswamy distribution as discussed in Arnold et al. [4] does allow correlation to vary over −1,1 but it has 5 parameters. In contrary, we propose an alternative (weighted) bivariate Kumaraswamy distribution that has 4 parameters and allows correlations over the full range −1,1.

Distribution of the Z = min(X, Y) and W = max(X, Y)

Suppose, we want to derive the distribution of Z=minX,Y and W=maxX,Y. Note that, for each z∈0<z<1, we have

PZ>z=∫z1∫zyfx,ydxdy+∫z1∫zxfx,ydxdy =C1−11−zab1+b2−b1+b2b0+b1+b21−zab0+b1+b2.

On differentiating PZ⩽z=1−PZ>z w.r.t. z, the density of Z will be

fZz=aC1−1za−1b1+b21−zab1+b2−1−1−zab0+b1+b2−1×I0<z<1.

Next, consider the distribution of W. For the distribution of W=maxX,Y, note that for any w∈0,1,

F¯Ww=PW>w =PX>w or Y>w =F¯Xw+F¯Yw−F¯Zw.

From (12), the corresponding density will be

fWw=fXw+fYw−fZw =aC1−1wa−1b0b1b0+b21−wab1−1+b0b2b0+b11−wab2−1  −b1+b21−wab1+b2−1−1−wab0+b1+b2−1b1b22b0+b1+b2b0+b1b0+b2  ×I0<w<1.

Reliability parameter: In this case we have from (6),

R=PY>X=∫01∫x∞fx,ydydx =b1b1+b2,
after some algebraic simplification.

3. TYPE II-WEIGHTED BIVARIATE KUMARASWAMY DISTRIBUTION

Let as before, W1,W2, and W0 be independent random variables with density functions fWiwi, i = 0, 1, 2. We consider the joint density of W1,W2 given that W0>max {W1,W2}. Now, if we define X,Y=d(W1,W2|max (W1,W2)<W0), then the density function of the corresponding bivariate weighted distribution is given by

fX,Yx,y=fW1xfW2yPmaxW1,W2<W0|W1=x,W2=yPmaxW1,W2<W0 =fW1xfW2yPW0>maxx,yPmaxW1,W2<W0 =fW1xfW2yF¯W0maxw1,w2PmaxW1,W2<W0.

Suppose that we have three independent Kumaraswamy variables. We need to evaluate PW0>maxW1,W2. Note that PW0>maxW1,W2=PW1<W2<W0+PW2<W1<W0.

Consider

P(W1<W2<W0)
=∫01∫w11∫w21a3b0b1b2w1a−1w2a−1w0a−11−w1ab1−11−w2ab2−11−w0ab0−1dw0dw2dw1=∫01∫w11a2b1b2w2a−11−w2ab0+b2−1dw2dw1=ab1b2b0+b2∫01w1a−11−w1ab0+b1+b2−1dw1=b1b2b0+b2b0+b1+b2.

Analogously, PW2<W1<W0=b1b2b0+b1b0+b1+b2, so that PW0>maxW1,W2=b1b22b0+b1+b2b0+b2b0+b1b0+b1+b2. For notational simplicity, let us write

D=PW0>maxW1,W2=b1b22b0+b1+b2b0+b2b0+b1b0+b1+b2.

Therefore,

fW0>max {W1,W2}.x,y
=D−1a2b1b2xya−11−xab1−11−yab2−11−max x,yab0=a2b1b22b0+b1+b2b0+b2b0+b1b0+b1+b2−1w1a−1w2a−1w0a−11−w1ab1−11−w2ab0+b2−1,0<x<y<1a2b1b22b0+b1+b2b0+b2b0+b1b0+b1+b2−1w1a−1w2a−1w0a−11−w1ab0+b1−11−w2ab2−1,0<y<x<1.

In this case the marginals are given by

fXx=2b0+b1+b2b0+b2b0+b1b0+b1+b2−1ab2−1xa−11−xab0+b1−11−b0b0+b21−xab2I0<x<1.
fYy=2b0+b1+b2b0+b2b0+b1b0+b1+b2−1ab1−1ya−11−yab0+b2−11−b0b0+b11−yab1I0<y<1.

Remark 1.

From (16) and (17), the marginal densities of Type-II weighted bivariate Kumaraswamy variable are linear combinations of univariate Kumaraswamy densities. In particular, using convenient transparent notation, one may write

fXx=b0+b2b0+b1+b2b22b0+b1+b2KWa,b0+b1−b0b0+b2b0+b1+b2b22b0+b1+b2KWa,b0+b1+b2,fXx=b0+b1b0+b1+b2b12b0+b1+b2KWa,b0+b2−b0b0+b1b0+b1+b2b12b0+b1+b2KWa,b0+b1+b2,
where KW(.,.) stands for the univariate Kumaraswamy distribution.

Distribution of the Z = min(X, Y) and W = max(X, Y)

In this case, following similar technique as before, the density of Z and W will be, respectively

fZz=ab0+b1+b2za−11−zab0+b1+b2−1×I0<z<1.
fWw=a2b0+b1+b2b0+b2b0+b1b0+b1+b2−1ab2−1wa−1b2−11−wab0+b1−1+b1−11−wab0+b2−1  −awa−1b0b0+b1+b2b0b2+b22+b0b1+b12b1b22b0+b1+b21−wab0+b1+b2−1×I0<w<1.

Note: It is interesting to see here that the distribution of Z is again a Kumaraswamy distribution with parameters a and b0+b1+b2.

Now, in this case, for any r⩾1, the marginal moments of X and Y are given by, respectively

EXr=b0+b2b0+b1+b2b22b0+b1+b2b0+b1Bb0+b1,ra  −b0b0+b2b0+b1+b2b22b0+b1+b2b0+b1+b2Bb0+b1+b2,ra,
and
EYr=b0+b1b0+b1+b2b12b0+b1+b2b0+b2Bb0+b2,ra  −b0b0+b1b0+b1+b2b12b0+b1+b2b0+b1+b2Bb0+b1+b2,ra.

The correlation coefficient ρ for this distribution is given by

ρ=M1M2M3,
where
M1=EXY−EXEY =∑k=0∞1/a+k−1k1b0+b1+b2+k  −b0+b2b0+b1+b2b22b0+b1+b2b0+b1Bb0+b1,1a−b0b0+b2b0+b1+b2b22b0+b1+b2b0+b1+b2Bb0+b1+b2,1a  ×b0+b1b0+b1+b2b12b0+b1+b2b0+b2Bb0+b2,ra−b0b0+b1b0+b1+b2b12b0+b1+b2b0+b1+b2Bb0+b1+b2,ra,
and
M2=VarX =b0+b2b0+b1+b2b22b0+b1+b2b0+b1Bb0+b1,2a−b0b0+b2b0+b1+b2b22b0+b1+b2b0+b1+b2Bb0+b1+b2,2a  −b0+b2b0+b1+b2b22b0+b1+b2b0+b2Bb0+b2,1a−b0b0+b2b0+b1+b2b22b0+b1+b2b0+b1+b2Bb0+b1+b2,1a2.

Similarly,

M3=VarY =b0+b1b0+b1+b2b12b0+b1+b2b0+b2Bb0+b2,2a−b0b0+b1b0+b1+b2b12b0+b1+b2b0+b1+b2Bb0+b1+b2,2a  −b0+b1b0+b1+b2b12b0+b1+b2b0+b2Bb0+b2,1a−b0b0+b1b0+b1+b2b12b0+b1+b2b0+b1+b2Bb0+b1+b2,1a2.

Remark 2.

We may write the following:

  1. Since maxW1,W2 is a TP2 function, the density corresponding to (15) will also be TP2. Furthermore, TP2 is the most rigid dependence property, several other dependency properties will follow immediately. Consequently, we can write the following:

    • X and Y are positive quadrant dependent.

    • XY is a positive regression dependent of YX.

    • XY is a left tail decreasing in YX.

  2. From the expression of the correlation coefficient, it can be conjectured that, like the Type-I BWK model, Type-II weighted bivariate Kumaraswamy model also allows the correlation coefficient between −1,1.

4. COPULA BASED BIVARIATE KUMARASWAMY DISTRIBUTION

In this section we consider two modified versions of the FGM (henceforth, in short) bivariate copula to construct a bivariate Kumaraswamy distribution. We list them as follows:

  • We begin by considering a modified version of bivariate FGM copula, given as

    Cu,v=uv1+θ1−uδ11−vδ2,
    for δ1,δ2>0 and θ∈−1,1. From now on, we call this as modified FGM copula based bivariate Kumaraswamy model (Type I), henceforth in short, Type-III bivariate Kumaraswamy model. Note that (18) is indeed a copula as it satisfies the following:

    • – C0,0=0; C1,1=1.

    • – C0,1=0=C1,0.

    • – For every u1⩽u2 and v1⩽v2, Cu2,v2−Cu2,v1−Cu1,v2+Cu1,v1⩾0.

      Next, suppose that Xi∼Kai,bi, for i=1,2 and they are independent. Then setting u=Fx1 and v=Fx2, a bivariate dependent Kumaraswamy model from (18) (hence forth Type-III bivariate Kumaraswamy) can be obtained as (the associated distribution function)

      HType−IIIx1,x2=1−1−x1a1b11−1−x2a2b21+θ1−x1a1b1δ11−x2a2b2δ2.

  • Another modified version of the FGM copula which can be used to construct a different bivariate Kumaraswamy model is given as follows:

    Cu,v=uδ1vδ21+θ1−uδ11−vδ2,
    with δ1,δ2>0, θ∈−1,1. One can easily show that (20) is also a valid copula. Again, we consider two arbitrary independent Xi∼Kai,bi, for i=1,2 and they are independent. Then, setting u=Fx1 and v=Fx2, a bivariate dependent Kumaraswamy model from (20) (hence forth Type-IV bivariate Kumaraswamy) can be obtained as (the associated distribution function)
    HType−IVx1,x2=1−1−xa1b1δ11−1−xa2b2δ2  1+θ1−1−1−x1a1b11−1−1−x2a2b2.

4.1. Properties of the Bivariate Copula Based Kumaraswamy Model

  • We provide the joint and the conditional copula density function expressions for one of the copula models (Type III) described earlier. For the Kumaraswamy copula (Type III), the corresponding joint copula density

    cu,v=∂2Cu,v∂u∂v =δ1δ2uδ1−1vδ2−12+θ1−uδ11−vδ2+θ1−uδ1−11−vδ2−1  +δ2uδ1vδ2−11−θδ11−uδ1−11−vδ2+δ1uδ1−1vδ21−θδ11−uδ11−vδ2−1.

    Also, the conditional copula density of U given V=v is given by

    cu|v=uδ1δ2vδ2−11+θ1−uδ11−vδ2+δ2uδ1vδ21−θ1−uδ11−vδ2−1.

    Similarly, one can get the other conditional copula density function. Following similar logic, one can get the corresponding density and conditional density function(s) for the Type-IV bivariate Kumaraswamy copula model.

  • Dependence structure Note that the selection of a particular copula function, indeed, depends on a number of factors, among which the dependence parameter is of primary importance. Furthermore, it is obvious from (19) and (21) that the resultant bivariate distributions reduce to the case of independence, when the parameter θ=0. These singular characteristics make these copulas particularly interesting for empirical analysis, as it is straightforward to compare estimates of the parameters of Fx,y and F1x and F2y separately. To study the nature of dependence, we consider the following two measures listed below:

    1. Kendall's τ: Let X and Y be continuous random variables with copula C. Then Kendall's τ is given by

      τX,Y=4∬[0,1]2Cu,vdCu,v−1=4∬[0,1]2cu,vdudv−1,
      where cu,v is the corresponding copula density.

    2. Spearman's ρ: Let X and Y be continuous random variables with copula C. Then Spearman's ρs is given by

      ρs=12∬[0,1]2uvdCu,v−3.

Based on the above, we can write the following:

  • For the bivariate Kumaraswamy (Type-III) copula

    1. Spearman's correlation coefficient ρ will be (assuming δ1 and δ2 are integers)

      ρ=121δ1δ2+θBδ1+1,δ1+1Bδ2+1,δ2+1−3.

    2. Kendall's τ will be (assuming δ1 and δ2 are integers)

      τ=4δ1δ22δ2+1Bδ1+1,δ1+3θBδ2+1,δ2+1Bδ1,δ1+1  +4δ1δ2θ2Bδ2+1,2δ2+1Bδ1,2δ1+1+θ2Bδ2+1,2δ2Bδ1,2δ1−1.

  • For the bivariate Kumaraswamy (Type-IV) copula

    1. Spearman's correlation coefficient ρ will be (assuming δ1 and δ2 are integers)

      ρ=121δ1+1δ2+1+θB2,1δ2+1B1,1δ1+1−3.

    2. Kendall's τ will be (assuming δ1 and δ2 are integers)

      τ=41δ1δ2+θ4B2,2+1δ1+2−B1,1δ1+1−1.

Next, we discuss the upper tail dependence and lower tail dependence property for these bivariate Kumaraswamy type copula models.

  • Tail dependence property: The upper tail dependence coefficient (parameter) λU is the limit (if it exists) of the conditional probability that Y is greater than 100α th percentile of G given that X is greater than the 100α th percentile of F as α approaches 1, λU=limα↑1PY>G−1α|X>F−1α. If λU>0, then X and Y are upper tail dependent and asymptotically independent otherwise. Similarly, the lower tail dependence coefficient is defined as λL=limα↓0PY⩽G−1α|X⩾F−1α. Let, C be the copula of X and Y. Then, equivalently we can write λL=limu↓0Cu,uu and λU=limu↓0C~u,uu, where C~u,u is the corresponding survival copula given by C~u,u=1−2u+Cu,u.

    For the bivariate Kumaraswamy (Type-III) copula, it is straightforward to see that λL=0, which implies that X and Y are asymptotically independent. Again, we have λU=0, thereby implying that X and Y are asymptotically dependent. In a similar way, one can establish these properties for the Type-IV bivariate Kumaraswamy type copula model.

  • Left-Tail decreasing property and Right-Tail increasing property Nelson [21] showed that XY is left tail decreasing, that is, LTDY|X and LTDX|Y if and only if for all u,u′,v,v′ such that 0<u⩽u′⩽1 and 0<v⩽v′⩽1, if Cu,vuv⩾Cu′,v′u′v′. Next, we have the following theorem:

Theorem 3.

The bivariate (Type-III) Kumaraswamy type copula in (18) has LTDY|X and LTDX|Y if and only if θ∈0,1 and for integer valued δ1 and δ2.

Proof.

Since, 0<u⩽u′⩽1 and 0<v⩽v′⩽1, we may write

1−uδ11−vδ2⩾1−u′δ11−v′δ2,
for any δ1>0, δ2>0 and both are integer valued. Next, for θ∈0,1, we can write
1+θ1−uδ11−vδ2⩾1+θ1−u′δ11−v′δ2.

Hence, Cu,vuv=1+θ1−uδ11−vδ2⩾1+θ1−u′δ11−v′δ2. This immediately implies the result.

Note that if θ∈−1,0, then the above tail dependence property will not hold. Almost identical argument will lead us to the fact that Left-Tail decreasing property and Right-Tail increasing property will also hold for bivariate (Type-IV) Kumaraswamy type copula.

Remark 3.

  • When δ1 and δ2 are not integers, the expressions for Kendalls τ and Spearmans ρ will involve infinite sums, but, still it will be in a closed form.

  • Since, in general, any convex combination of two (or more) is again a copula, one might consider another bivariate Kumaraswamy type copula (say, Type-V) with the following structure: CType−Vu,v=βCType−IIIu,v+1−βCType−IIu,v, for suitable β∈0,1. For a detailed study on the Arnold–Ng type bivariate copula based Kumaraswamy distribution construction and other associated bivariate copula models, see, Arnold and Ghosh [12].

5. MULTIVARIATE WEIGHTED KUMARASWAMY DISTRIBUTION

Following Arnold et al. [13], we consider the model in which in which T1,T2,…,Tj are i.i.d. random variables with distribution and density functions G0 and g0; X1,X2,…,Xk are i.i.d. random variables with distribution and density functions F0 and f0 and U1,U2,…,Uℓ are i.i.d. random variables with distribution and density functions H0 and h0. In this case we have

fx1,x2,…,xk∝∏β=1kf0xβG0x1:kj1−H0xk:kℓ.

When the three distributions in (22) are of the Kumaraswamy form, it reduces to

fx1,x2,…,xk∝akbk∏i=1kxia−11−xiab−11−1−x1:kab0j1−xk:kaℓb1.

To identify the required normalizing constant we must evaluate

∫01∫01⋯∫01akbk∏i=1kxia−11−xiab−11−1−x1:kab0j1−xk:kaℓb1dx1⋯dxk=∑m=0jjm−1mE1+X1:kamb11+Xk:ka−ℓ b0,
where the Xi's have a K a,b distribution. Next, the joint distribution of X1:k and Xk:k for a random sample of size k will be
fx1:k,xk:k=kk−1a2b0b1x1:kaxk:ka1−x1:kab01−xk:kab1I0<x1:k<xk:k<1.

From (24) and on using (25), the normalizing constant is

C=∑m=0jjm−1mkk−1b0b0+mb1+1b01+ℓ+b11+m+b0+2.

Hence, the k-variate joint density function in (25) can be written as

fx1,x2,…,xk=C−1akbk∏i=1kxia−11−xiab−11−1−x1:kab0j1−xk:kaℓb1  ×I0<x1,x2,…,xk<1.

Another multivariate extension of the BWK model can be proposed using (3) as follows

fx1,x2,…,xk=D−1∑i=0k∏j=1kakbj1−1−x1:kab0I0_<x_<1_,
where D=b0∑j=0kbi.

6. MAXIMUM LIKELIHOOD ESTIMATION

In this section, we consider the estimation of the model parameters of the BWK distribution.

Suppose we have n observations from the bivariate density in (6). The log-likelihood is given by

ℓa,b0,b1,b2=−nlog b0b0+b1+b2+2nlog a+2nlog b1+log b2  +a−1∑i=1nlog xi+log yi  +b1−1∑i=1nlog 1−xia+b2−1∑i=1nlog 1−yia  +∑i=1nlog 1−1−min (xi,yi)ab0.

The corresponding likelihood equations are

∂ℓ∂b0=∑i=1n−1−1−minxi,yiab0log 1−1−minxi,yia1−1−1−minxi,yiab0  −b0+b1+b21b0+b1+b2−b0b0+b1+b22nb0.
∂ℓ∂b1=∑i=1nlog 1−xia+2nb1+nb0+b1+b2.
∂ℓ∂b2=∑i=1nlog 1−yia+2nb2+nb0+b1+b2.
∂ℓ∂a=∑i=1nb01−minxi,yialog1−minxi,yi1−1−minxi,yiab0−11−1−1−minxi,yiab0+2na  +b1−1∑i=1n−xialogxi1−xia+b2−1∑i=1n−yialogyi1−yia+∑i=1nlogxi+∑i=1nlogyi.

Setting (27–30) to 0 and solving these likelihood equations simultaneously, we get the maximum likelihood estimates (MLEs) for b0, b1, b2, and a.

7. APPLICATION

In this section we consider two applications of the proposed BWK distribution based two data sets:

  • Data Set I: Earthquakes become major societal risks when they strike on vulnerable populations. We consider the data is obtained from Ozel [18]. Due to the fact that a significant portion of Turkey is subject to frequent earthquakes, destructive mainshocks and their foreshock and aftershock sequences between the longitudes (39 − 42°N) and latitudes 26−45°E are investigated. In this particular region, 111 mainshocks with surface magnitude Ms of five or more have occurred in the past 106 years. We define the following random variables: X represents the magnitude of foreshocks and Y represents the magnitude of the aftershocks. We fit the data to the following bivariate Kumaraswamy models:

  • Data Set II: The data on 37 patients were available regarding the hemoglobin content in blood being prone to type II diabetes from a Private Clinic in Tennessee. To see the effect of reducing hemoglobin content in the blood a special type of treatment was administered to those patients. We define the following: X as a random variable which represents the proportion of hemoglobin content in the blood before the treatment, Y as a random variable which represents the proportion of hemoglobin content in the blood after the treatment.

    1. Model I: BWK distribution ([6]).

    2. Model II: Bivariate Kumaraswamy (absolutely continuous distribution (Wagner et al. [5], [5]).

    3. Model III: Bivariate Kumaraswamy distribution via conditional specification (Arnold and Ghosh [12]), given by

      fx,y=Cα1α2xα1−1yα2−11−xα1β1−11−yα2β2−1expβ3log1−xα1log 1−xα2  ×I0<x<1,0<y<1,
      where C is an appropriate normalizing constant.

    4. Model IV: Bivariate Kumaraswamy distribution via conditional survival specification (Arnold and Ghosh [4]), given by

      fx,y=α1α2x1α1−1yα2−11−xα1β1−11−yα2β2−1expβ3log1−x1α1log1−yα2  ×β1β2+β2β3log1−x1α1+β3+β32log1−xα1log1−yα2+β1β3log1−yα2I0<x1,x2<1.

    5. Model V: Nadarajah [19] bivariate F3-beta distribution, given by

      fx,y=Cxβ−1yδ−11−x−yγ−β−δ−11−uxθ11−vyθ2,
      for 0<x<1, 0<y<1, 0<x+y<1, −1<u<1, −1<v<1, β,δ,θ1,θ2>0 and γ>β+δ, and C is the normalizing constant.

    6. Model VI: Nadarajah [3] bivariate generalized beta distribution given by

      fx,y=Cxα−1yβ−11−xγ−α−11−yγ−β−11−xyδγ,
      for 0<x<1, 0<y<1, 0<x+y<1, −1<u<1, −1<v<1, β,δ,θ1,θ2>0 and γ > β + δ, and C is the normalizing constant.

    7. Model VII: Olkin and Trikalinos [20] bivariate beta distribution given by

      fx,y=xα1−1yα2−11−xα0+α2−11−yα0+α1−11−xyα0+α1+α2.

To check the goodness of fit of all statistical models, several other goodness-of-fit statistics are used and are computed using computational package Mathematica. The MLEs are computed using Nmaximize technique as well as the measures of goodness-of-fit statistics including the log-likelihood function evaluated at the MLEs. Parameter estimates along with several goodness of fit measures are provided in Tables 1 and 3 and in Tables 2 and 4 for the data sets I and II, respectively.

Model Model I Model II Model III Model IV
Parameter estimates a^=2.49560.8010 α1^=3.52870.3335 α1^=2.45710.4620 α1^=1.18920.6043
b1^=1.48910.1207 α2^=1.18450.9723 α2^=2.51830.5781 α2^=2.03670.1472
b2^=0.31650.0842 α3^=3.24240.1065 β1^=1.4720.1368 β1^=4.3520.4956
b0^=0.78490.0229 β^=1.9230.9537 β2^=1.18830.0442 β2^=2.17690.0154
β3^=1.65430.0279 β3^=2.5380.4015
Log-likelihood −162.34 −116.58 −113.36 −113.25
χ2 goodness p-value 0.9983 0.7146 0.8345 0.9978
Table 1

Parameter estimates for Data Set I.

Model Model I Model II Model III Model IV
Parameter estimates a^=1.27810.3533 α1^=2.12580.4015 α1^=0.5470.6436 α1^=0.7230.5145
b1^=1.11390.0737 α2^=3.47530.1216 α2^=1.67820.1635 α2^=1.78940.2051
b2^=1.14640.2102 α3^=2.38320.3528 β1^=0.35870.1822 β1^=0.22910.0567
b0^=0.87350.1718 β^=1.29130.4583 β2^=0.71170.3453 β2^=0.56730.1644
β3^=0.32570.5726 β3^=0.48820.8293
Log-likelihood −162.34 −116.58 −113.36 −113.25
χ2p-value 0.9532 0.8140 0.9123 0.9203
Table 2

Parameter estimates for Data Set II.

Model Model V Model VI Model VII
Parameter estimates β^=2.4270.6518 α^=1.4850.6672 α0^=4.0160.6436
δ^=0.9802.490 β^=1.9730.3294 α1^=3.76492.1873
γ^=1.08924.5253 γ^=7.5280.9897 α2^=6.1720.5837
θ1^=2.68430.6439 δ^=2.7350.9892
θ2^=0.87350.1718
Log-likelihood −245.68 −237.18 −186.79
χ2p-value 0.5584 0.6283 0.7246
Table 3

Parameter estimates for Data Set I.

Model Model V Model VI Model VII
Parameter estimates β^=2.8930.753 α^=0.8911.248 α0^=3.1840.776
δ^=1.6531.046 β^=1.1721.682 α1^=2.4571.2837
γ^=3.1173.8931 γ^=6.4181.0583 α2^=3.2710.4819
θ1^=1.5420.8568 δ^=1.3892.4015
θ2^=0.4890.3294
Log-likelihood −262.17 −206.44 −135.43
χ2p-value 0.3518 0.3849 0.6722
Table 4

Parameter estimates for Data Set II.

8. CONCLUSION

In recent times the construction of bivariate and multivariate Kumaraswamy distributions has received a significant amount of attention. While most of the other works focuses primarily on investigating structural properties of the proposed model, in our present work, we try to provide more emphasize on the application side without undermining the need to discuss structural properties of the developed bivariate and multivariate Kumaraswamy distributions. In this article, we propose a 4 parameter bivariate (weighted) Kumaraswamy distribution, which allows the correlation coefficient to vary over the full range −1,1 and it's an improved model in comparison with other bivariate Kumaraswamy type distributions (such as those studied and discussed in Arnold et al. [14], [15] with 5 parameters), since we have one parameter less. Thus it merits a separate study. In conclusion, the BWK distributions provides a rather flexible mechanism for fitting a wide spectrum of positive real world data. Additionally, one can easily imagine situations in which observations are made only if the maximum of k variables is less than one particular variable. In such a scenario, within the framework of a multivariate Kumaraswamy joint distribution for the k+1-dimensional data, efforts will be made. Indeed, one could begin with any one of the many dependent multivariate Kumaraswamy models available in the literature. A separate report on such models will be prepared. However, unless k is small, these models necessarily involve a considerable number of parameters, which can be expected to invite difficulties in estimation in practical settings where sample sizes cannot be expected to be enormous. Furthermore one can envision a semi-parametric model in which X,Y=h1X*,h2Y*, where X*,Y* has a BWK distribution and h1 and h2 are unknown functions to be estimated from the data. Alternatively the functions h1 and h2 might be assumed to belong to specific parametric families of functions. The enhanced flexibility of such augmented models may prove to be useful in many applications.

ACKNOWLEDGMENTS

The author is grateful for some invaluable comments made by two anonymous referees which has greatly helped to improve on an earlier version of this manuscript. The author has no conflict of interest. Furthermore, this research received no specific grant from any funding agency in the public, commercial, or not-for-profit sector.

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7.C. Gupta and P. Keating, Scand. J. Stat., Vol. 13, 1985, pp. 49-56.
12.B.C. Arnold and I. Ghosh, J. Appl. Stat. Sci., Vol. 22, 2016, pp. 37-60.
13.B.C. Arnold and I. Ghosh, RevStat, Vol. 15, 2016, pp. 223-250.
15.C.R. Rao, Sankhyā: Indian J. Stat. Ser. A, Vol. 27, 1965, pp. 311-324.
16.G.P. Patil and C.R. Rao, P.R. Krishnaiah (editor), Applications in Statistics, North-Holland, Amsterdam, 1977, pp. 383-405.
17.M. Mahfoud and G.P. Patil, G. Kallianpur et al. (editors), Statistics and Probability Essays in Honor of C.R. Rao, North-Holland, Amsterdam, 1982, pp. 479-492.
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Journal
Journal of Statistical Theory and Applications
Volume-Issue
18 - 3
Pages
198 - 211
Publication Date
2019/09/16
ISSN (Online)
2214-1766
ISSN (Print)
1538-7887
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10.2991/jsta.d.190619.001How to use a DOI?
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risenwbib
TY  - JOUR
AU  - Indranil Ghosh
PY  - 2019
DA  - 2019/09/16
TI  - Bivariate and Multivariate Weighted Kumaraswamy Distributions: Theory and Applications
JO  - Journal of Statistical Theory and Applications
SP  - 198
EP  - 211
VL  - 18
IS  - 3
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.190619.001
DO  - 10.2991/jsta.d.190619.001
ID  - Ghosh2019
ER  -