# Bivariate Mixture of Inverse Weibull Distribution: Properties and Estimation

Authors: A. S. AL-Moisheer; Refah Mohammed Alotaibi; Ghadah A. Alomani; H. Rezk
Journal: Mathematical Problems in Engineering
 (2020)

---

## Abstract

In this study, we construct a mixture of bivariate inverse Weibull distribution. We assumed that the parameters of two marginals have Bernoulli distributions. Several properties of the proposed model are obtained, such as probability marginal density function, probability marginal cumulative function, the product moment, the moment of the two variablesx and y, the joint moment-generating function, and the correlation between x and y. The real dataset has been analyzed. We observed that the mixture bivariate inverse Weibull distribution provides a better fit than the other model.


---

## Body

## 1. Introduction

In the history of statistics, the use of finite mixture models is very old. It was particularly used to model population heterogeneity and generalize distributional assumptions, clustering and classification, and so on. Eight years later, Pearson [1] considered a mixture of two univariate Gaussian distributions to estimate the parameters of the model using the method of moments (MOM) to analyse a dataset containing ratios of forehead to body lengths for 1,000 crabs. Since then several authors have studied finite mixture models under different scenarios. Mendenhall and Hader [2] considered exponentially distributed failure time distributions based on censored lifetime data to estimate the model parameters using the maximum likelihood method. In their study, they divided the failure population into two subpopulations, each representing a different cause or type of failure. Radhakrishna et al. [3] in their paper considered both moment and maximum likelihood estimators of the unknown parameters of two-component mixture of the generalized gamma distribution. Ahmed et al. [4] obtained approximate Bayes estimators for parameters of mixture of two Weibull distributions under type-II censoring.Inverse Weibull (IW) distribution has been used quite successfully to analyze lifetime data which has a nonmonotone hazard function. The IW distribution can be readily applied to a wide range of situations including applications in medicine, reliability, and ecology. Keller et al. [5] obtained the IW model by investigating failures of mechanical components subject to degradation phenomena such as the dynamic components of diesel engines; see, for example, Murthy et al. [6]. The physical failure process given by Erto and Rapone [7] also leads to the IW model. Erto and Rapone [7] showed that the IW model provides a good fit to survival data such as the times to breakdown of an insulating fluid subject to the action of constant tension, see [8]; Calabria and Pulcini [9] provided an interpretation of the IW distribution in the context of a load strength relationship for a component.There have been several attempts made in the last few years to introduce bivariate distribution. For example, Sarhan and Balakrishnan [10] suggested a bivariate distribution that is more flexible than the bivariate exponential distribution. Later, this distribution was modified by Kundu and Gupta [11]. Kundu and Gupta [11, 12] introduced the bivariate generalized exponential and bivariate proportional reversed hazard distributions, respectively, and argued its different properties. Sarhan et al. [13] proposed bivariate generalized linear failure rate distributions and discussed its several properties. Sarabia et al. [14] proposed three new classes of bivariate beta-generated distributions, and these classes were created using three alternative definitions of bivariate distributions with classical beta marginals and different covariance structures. Muhammed [15] proposed the bivariate inverse Weibull distribution, and Al-Mutairi et al. [16] introduced bivariate and multivariate weighted Weibull distribution distributions with weighted Weibull marginals and established their several properties.The mixture of inverse Weibull distribution has also been found useful in some applications. For example, Al Moisheer [17] used an inverse Weibull mixture distribution in measuring the carbon monoxide level in different locations of the Jeddah city. Theoretical work on mixture inverse Weibull was introduced by Sultan et al. [18]; they discussed some properties of the mixture of the inverse Weibull model with some graphs of the density and hazard function. Sultan and Moisheer [19] found the maximum likelihood estimates of the parameters of the mixture of two inverse Weibull distributions by using classified and unclassified observations. The notion of mixing may be regarded as a special case of compounding. In the bivariate case, let X and Y be two random variables with parameters θ1 and θ2, respectively. For given fixed values of (θ1, θ2), X and Y may or may not be independent. The basic idea of compounding is to say that θ1 and θ2 are themselves random variables which are not constant, and the observed (marginal) distribution of X and Y results from integrating over the joint distribution of θ1 and θ2, i.e.,(1)hx,y=∬hx,y,θ1⋅θ2dθ1dθ2.The primary objective of the paper is construct a bivariate of inverse Weibull mixture distribution by assuming two independent inverse Weibull distributions with the scale parameters having a generalized bivariate Bernoulli distribution.The organization of the paper is as follows. The description of the proposed mixture of bivariate inverse Weibull distribution along with basic properties is reported in Section2. In Sections 2 and 3, we give the moment and correlation definition for the bivariate mixture inverse Weibull model. In Section 5, we use the EM algorithm method. The application study is carried out in Section 6. Finally, concluding remarks are presented in Section 7.


## 2. Mixture of Bivariate Inverse Weibull Distribution

A random variable with an inverse Weibull distribution (BIWM) has the cumulative distribution function (cdf) and the probability density function (pdf) in the following from:(2)Fx,θ,λ=e−λx−θ,x>0,(3)fx,θ,λ=θλx−θ−1e−λx−θ,x>0,where θ > 0 and λ > 0 are the shape and scale parameters.In the bivariate case, letX and Y be two random variables with parameters θ1 and θ2, respectively. For given fixed values of θ1, θ2, X and Y are independent.The probability density function of bivariate inverse Weibull mixture distribution is given as(4)Fx,θi=∑i=12pifixi,θi,0≤pi≤0,p1+p2=1,where pdf of the first component of inverse Weibull is given by equation (3) with fixed shape parameter θ > 0 and random scale parameter λ > 0 taking two distinct values λ1 and λ2. Similarly for fixed shape parameter θ2, let Y have an inverse Weibull mixture density and the pdf of the second component (inverse Weibull) is given by(5)gy,φ,β=φβy−φ−1e−βy−φ,y>0,with β being a random scale parameter taking values β1 and β2.For given values (λ, β), we assume that X and Y are independent, but λ and β are correlated through their generalized bivariate distribution with the following probability matrix:(6)Leth(x, y) be the joint probability density function of (X, Y), then(7)hx,y=fx|θ,λ1gy|φ,β1pλ1β1+fx|θ,λ1gy|φ,β2pλ1β2+fx|θ,λ2gy|φ,β1pλ2β1+fx|θ,λ2gy|φ,β2pλ2β2=θλ1x−θ−1e−λ1x−θφβ1y−φ−1e−β1y−φpλ1β1+θλ1x−θ−1e−λ1x−θφβ2y−φ−1e−β2y−φpλ1β2+θλ2x−θ−1e−λ2x−θφβ1y−φ−1e−β1y−φpλ2β1+θλ2x−θ−1e−λ2x−θφβ2y−φ−1e−β2y−φpλ2β2.Suppose that(8)a=pλ1β1,b=pλ1β2,c=pλ2β1,d=pλ2β2=1−a−b−c.Substitutinga, b, c, and d in equation (7), we obtain(9)hx,y=θφx−θ−1y−φ−1λ1β1ae−λ1x−θ+β1y−φ+λ1β2be−λ1x−θ+β2y−φ+λ2β1ce−λ2x−θ+β1y−φ+λ2β2de−λ2x−θ+β1y−φ.The model can be seen to be capable of producing a wide variety of bivariate shapes. The joint density can be seen to be a mixture of four bivariate inverse Weibull distributions, requiring a total of nine parameters for its specification. For modeling purposes, it may be found that not all of the four components are required, so we may want to consider restrictions such asb = c = 0, a = d = 0, or a = b = c = 0, corresponding to correlations between the scale parameters of +1, −1, and 0, respectively. In Figure 1, we show some examples of contour plots for various values of a, b, c, and d with the shape and scale parameters fixed at θ = 8, φ = 3, λ1 = 0.25, λ2 = 0.4, β1 = 1.1, and β1 = 6.2.Figure 1
Contours of mixture of the bivariate inverse Weibull model withθ = 8, φ, = 3, λ1 = 0.25, λ2 = 0.4, β1, = 1.1, β1 = 6.2, and selected choices of the mixing parameters.
(a)
(b)
(c)
(d)
(e)
(f)The marginal densities ofX and Y, respectively, are(10)hXx=π1f1x+1−π1f2x,where π1 = a + b, and(11)hYy=π2g1y+1−π2g2y,where π2 = a + c.The cumulative distribution function is given by(12)Fx,y=∫0y∫0xht,sdtds=∫0y∫0xθφt−θ−1s−φ−1λ1β1ae−λ1t−θ+β1s−φdtds+∫0y∫0xθφt−θ−1s−φ−1λ1β2be−λ1t−θ+β2s−φdtds+∫0y∫0xθφt−θ−1s−φ−1λ2β1ce−λ2t−θ+β1s−φdtds+∫0y∫0xθφt−θ−1s−φ−1λ2β2de−λ2t−θ+β2s−φdtds=ae−λ1x−θ−1.e−β1y−φ−1+be−λ1x−θ−1.e−β2y−φ−1+ce−λ2x−θ−1.e−β1y−φ−1+de−λ2x−θ−1.e−β2y−φ−1.The survival function of BIWM distribution is given by(13)Rx,y=1−Fx,y.So,(14)Rx,y=1−ae−λ1x−θ−1.e−β1y−φ−1−be−λ1x−θ−1.e−β2y−φ−1−ce−λ2x−θ−1.e−β1y−φ−1−de−λ2x−θ−1.e−β2y−φ−1.The hazard rate function (hrf) is given as follows:(15)hrfx,y=θφx−θ−1y−φ−1λ1β1ae−λ1x−θ+β1y−φ+λ1β2be−λ1x−θ+β2y−φ+λ2β1ce−λ2x−θ+β1y−φ+λ2β2de−λ2x−θ+β1y−φ1−ae−λ1x−θ−1⋅e−β1y−φ−1−be−λ1x−θ−1.e−β2y−φ−1−ce−λ2x−θ−1.e−β1y−φ−1−de−λ2x−θ−1.e−β2y−φ−1.The conditional probability function ofx for given y is given by(16)hx|y=hx,yhy,x>0=θφx−θ−1y−φ−1λ1β1ae−λ1x−θ+β1y−φ+λ1β2be−λ1x−θ+β2y−φ+λ2β1ce−λ2x−θ+β1y−φ+λ2β2de−λ2x−θ+β1y−φπ2g1y+1−π2g2y,and the conditional probability function of y for given x is given by(17)hy|x=hx,yhx,x>0=θφx−θ−1y−φ−1λ1β1ae−λ1x−θ+β1y−φ+λ1β2be−λ1x−θ+β2y−φ+λ2β1ce−λ2x−θ+β1y−φ+λ2β2de−λ2x−θ+β1y−φπ1f1x+1−π1f2x.


## 3. Moments

In this section, we derive the statistical properties of BIWM distribution. The product moments about zero is given by(18)μij′=∫0∞∫0∞xyhx,ydxdy=a∫0∞θφx−θλ1e−λ1x−θdx⋅∫0∞β1y−φe−β1y−φdy+b∫0∞θφx−θλ1e−λ1x−θdx.∫0∞β2y−φe−β2y−φdy+c∫0∞θφx−θλ2e−λ2x−θdx.∫0∞β1y−φe−β1y−φdy+d∫0∞θφx−θλ2e−λ2x−θdx.∫0∞β2y−φe−β2y−φdy.Letz = λ1x−θ, w = β1y−φ, and dz = −θλ1x−θ−1 dx, at x = 0, z = 0, x = ∞, z = ∞, y = 0, w = 0, y = ∞, and w = ∞; hence, we obtain(19)μij′=a∫0∞z−1/θλ11/θe−zdz⋅∫0∞w−1/φβ1−1/φe−wdw+b∫0∞z−1/θλ11/θe−zdz⋅∫0∞w−1/φβ2−1/φe−wdw+c∫0∞z−1/θλ21/θe−zdz⋅∫0∞w−1/φβ1−1/φe−wdw+d∫0∞z−1/θλ21/θe−zdz⋅∫0∞w−1/φβ2−1/φe−wdw,and then(20)μij′=Γθ−1θΓφ−1φaβ11/φλ11/θ+bβ21/φλ11/θ+cβ11/φλ21/θ+dβ21/φλ21/θ.The expected values ofX and Y are(21)EX=−λ12+1/θΓ−1+θθ,−1≤θ≤0,λ1>0,EY=−β12+1/φΓ−1+φφ,−1≤φ≤0,β1>0.The joint moment-generating function of bivariate mixture is given by(22)μxys,t=∫0∞∫0∞etx+syhx,ydxdy=∫0∞∫0∞etx+syx−θ−1y−φ−1λ1β1ae−λ1x−θ+β1y−φ+λ1β2be−λ1x−θ+β2y−φ+λ2β1ce−λ2x−θ+β1y−φ+λ2β2de−λ2x−θ+β1y−φdxdy,μxys,t=a∫0∞λ1θetxx−θ−1e−λ1x−θdx⋅∫0∞β1φesyy−φ−1e−β1y−φdy+b∫0∞λ1θetxx−θ−1e−λ1x−θdx⋅∫0∞φβ2esyy−φ−1e−β2y−φdy+c∫0∞λ2θetxx−θ−1e−λ2x−θdx⋅∫0∞φβ1esyy−φ−1e−β1y−φdy+d∫0∞λ2θetxx−θ−1e−λ2x−θdx⋅∫0∞φβ2esyy−φ−1e−β2y−φdy.By using exponential expansion,(23)etx=∑i=0∞txii!,e−λ1x−θ=∑j=0∞−λ1x−θjj!,esy=∑k=0∞sykk!,e−β1y−φ=∑m=0∞−β1y−φmm!.Then, we have(24)μxys,t=a∫0∞λ1θ∑i=0∞txii!x−θ−1∑j=0∞−λ1x−θjj!dx⋅∫0∞β1φ∑k=0∞sykk!y−φ−1∑m=0∞−β1y−φmm!dy+b∫0∞λ1θ∑i=0∞txii!x−θ−1∑j=0∞−λ1x−θjj!dx⋅∫0∞β2φ∑k=0∞sykk!y−φ−1∑m=0∞−β2y−φmm!dy+c∫0∞λ2θ∑i=0∞txii!x−θ−1∑j=0∞−λ2x−θjj!dx⋅∫0∞β2φ∑j=0∞sykk!y−φ−1∑m=0∞−β2y−φmm!dy+d∫0∞λ2θ∑i=0∞txii!x−θ−1∑j=0∞−λ2x−θjj!dx⋅∫0∞β2φ∑k=0∞sykk!y−φ−1∑m=0∞−β2y−φmm!dy,and then(25)=aλ1θ∑i=0∞tii!∑j=0∞−λ1jj!∫0∞xi−θ−1−iθdx⋅β1φ∑k=0∞skk!∑m=0∞−β1mm!∫0∞yj−φ−1−jφdy+bλ1θ∑i=0∞tii!∑j=0∞−λ1jj!∫0∞xi−θ−1−iθdx⋅β2φ∑k=0∞skk!∑m=0∞−β2mm!∫0∞yj−φ−1−jφdy+cλ1θ∑i=0∞tii!∑j=0∞−λ2jj!∫0∞xi−θ−1−iθdx⋅β1φ∑k=0∞skk!∑m=0∞−β1mm!∫0∞yj−φ−1−jφdy+dλ2θ∑i=0∞tii!∑j=0∞−λ2jj!∫0∞xi−θ−1−iθdx⋅β2φ∑k=0∞skk!∑m=0∞−β2mm!∫0∞yj−φ−1−jφdy.Finally, we get(26)μxys,t=β1e−β1s−φaλ1e−λ1t−θ+bλ2e−λ2t−θ−β2e−β2s−φcλ1e−λ1t−θ+dλ2e−λ2t−θ.


## 4. Correlation

The covariance betweenX and Y for the BIWM model is given by(27)CovX,Y=EXY−EXEY.So,(28)CovXY=Γθ−1θΓφ−1φaβ11/φλ11/θ+bβ21/φλ11/θ+cβ11/φλ21/θ+dβ21/φλ21/θ−Γ−θ−1θΓ−φ−1φβ11/φ+2λ11/θ+2.It is noted that the covariance between two variablesX and Y depends only on the parameters, θ, φ, λ1β1, λ2β2, a, b, c, and d. The correlation coefficient of X and Y is given as follows:(29)ρXY=CovXYσXσY,where σX and σY are the standard deviation for X and Y, respectively. Then, the variance of X and Y are shown as follows:(30)σX2=EX2−EX2,σY2=EY2−EY2=λ12/θ+2Γ−θ−2θ−Γ−θ−1θλ11/θ+22,θ>2,λ1>0,σY2=β11/φ+2Γ−φ−2φ−Γ−φ−1θφβ11/φ+22,φ>2,β1>0,where(31)EX2=λ12/θ+2Γ−θ−2θ,EY2=β12/φ+2Γ−φ−2φ.Then, we can obtain the standard deviation forX and Y by taking the positive squared root of σX2 and σY2, respectively, as follows:(32)σX=σX2,σY=σY2.We compute numerically the correlation coefficientρXY. The numerical results are listed in Table 1 for different values of the parameters.Table 1
Correlation based on different values of parameters.
θ
φ
β
1
β
2
λ
1
λ
2
a
b
c
ρ
X
Y
0.2
0.2
1.25
1.5
0.5
2
0.4
0.3
0.3
0.0241
0.4
1.4
1.5
1
0.3
0.75
0.3
0.3
0.3
0.5793
0.6
1.6
0.25
0.3
0.6
0.5
0.4
0.4
0.1
0.8213
0.8
1.8
0.5
1.2
0.8
1.5
0.3
0
0.5
0.6973
1.2
2
0.8
1.7
1
1.2
0.5
0.3
0
0.1383
1.4
0.3
0.4
0.2
1.2
1.5
0.5
0
0.5
0.0974
1.6
0.9
1.1
0.5
1.5
0.1
0
0.5
0.3
0.8372
1.8
0.8
0.9
0.8
0.2
0.9
0.4
0.2
0.1
0.4739
2
0.5
0.2
2.5
1.7
1.3
0.2
0.3
0.4
0.3796


## 5. EM Algorithm

McLachlan and Krishnan [20] introduced the EM algorithm as a method of estimation; to apply the EM algorithm, we augment the data (xk, yk), k = 1, …, n with the group membership variables (ak, bk, ck), k = 1, …, n, where ak is one if the kth observation is in f(X, λ1, β1) and zero otherwise. Similarly, for bk, ck, we have four groups Gij, i, j = 1, 2, for which the densities are(33)fijX,Y=fiXfjY=θλix−θ−1e−λix−θφβjy−φ−1e−βjy−φ.The mixing proportions arePG11=a,PG12=b,PG21=c, and PG22=1−a−b−c.We defineℓijx,y=logfijx,y; then, the EM algorithm as a method of estimation is given by finding the complete log-likelihood ℓ as follows:(34)ℓ=∑k=1nakℓ11xk,yk+∑k=1nbkℓ12xk,yk+∑k=1nckℓ21xk,yk+∑k=1n1−ak−bk−ckℓ22xk,yk.These group membership variables (ak, bk, ck) so in the E step have a linear relation, so θ^,φ^,λ^1,λ^2,β^1,β^2,a,b,c are calculated as(35)a^k=a^f11xk,yka^f11xk,yk+b^f12xk,yk+c^f21xk,yk+1−a^−b^−c^f22xk,yk,and similarly for bk, ck. It should be noted that algebraic simplification of the above equation may be necessary to avoid numerical problems. For the M step, we need to maximize (34) over (θ, φ, β1, β2, λ1, λ2) for fixed values of (αk, bk, ck); this is expedited by the conditional independence of x and y given the group membership. We can essentially deal with the univariate separately. Differentiating (34) gives(36)∂ι∂λ1=nλ1∑k=1nak+bk−∑k=1nak+bkxk−θ,(37)∂ι∂λ2=nλ2∑k=1nck+dk−∑k=1nck+dkxk−θ,(38)∂ι∂β1=nβ1∑k=1nak+ck−∑k=1nak+ckyk−φ,(39)∂ι∂β2=nβ2∑k=1nbk+dk−∑k=1nbk+dkyk−φ,(40)∂ι∂θ=4nθ−4∑k=1nlogxk−2∑k=1nλ1xk−θlogxk−2∑k=1nλ2xk−θlogxk,(41)∂ι∂φ=4nφ−4∑k=1nlogyk−2∑k=1nβ1yk−φlogyk−2∑k=1nβ2yk−φlogφk.The M step is completed by setting(42)a^=1n∑k=1nak.We equate the systems of equations (36)–(41) to zero. We use the mixtools R package to solve these equations numerically. After the maximum likelihood estimators for λ1, λ2, β1, β2, θ, and φ are obtained, substitute these estimates in (ak, bk, ck). We complete the M step by setting a^=1/n∑k=1nak and so on.Initial values for the mixing proportions are obtained by using the moment’s method of the marginal univariate inverse Weibull mixtures, see [3, 21]. Then, we take the resulting estimates of the inverse Weibull parameters as starting values for the EM algorithm. After that, we merge the moment estimators of the marginal mixing parameters to obtain initial values for the bivariate mixing parameters, assuming the independence between two variables X and Y. We apply this method in application as mentioned in Section 7, specifically, in Table 2.Table 2
The parameter estimates from the BIWM model.
Estimates from BIWM
λ
^
1 = 18.89
λ
^
2
=
12.90
β
^
1
=
32.47
β
^
2
=
23.73
θ
^
=
31.76
φ
^
=
0.30
â = 0.15
b
^
=
0.35
c
^
=
0.20
d
^
=
0.30


## 6. Application

In this section, we use the data of cows, i.e., the data concerning the milk yield of dairy cows to illustrate the flexibility of the new model. Cow data are introduced by Tocher [22]. In Figure 2, we plot the marginal and joint contours, histogram cow data fitted for bivariate mixture of inverse Weibull distribution. We compare the fits of the new bivariate inverse Weibull mixture with the other competitive models, such as BIWM, bivariate gamma mixture (MBG), mixture inverse Weibull (MIW), and bivariate inverse Weibull (BIW) distributions. The comparison is done based on some measures of goodness of fit, the maximized log-likelihood under the −ℓ^, Akaike information criterion (AIC), Bayesian information criterion (BIC), consistent Akaike information criterion (CAIC), Hannan–Quinn information criterion (HQIC), and Kolmogorov–Smirnov (KS) statistic and its P value (PV). Cow data represent age and lactation period of cows. The first variable X is age (years), and the second variableY is lactation period (weeks). We obtain the maximum likelihood estimators and the corresponding log values for the inverse Weibull mixture distribution of the cow data using the EM algorithm. The examination of the marginal density mixture f(x) of the model is given in Table 3. The examination of the marginal density f(y) of the model is given Table 4. Figures 3–6 give the estimated CDF (E-CDF), the estimated density (E-PDF), P-P plot (P-P), and Q-Q plot for f(x) or f(y). Now, we fit the data to MBIW, BGM, MIW, and BIW distributions. It is noted that the proposed model fits the cow data better than the BGM, MIV, and BIW distributions. Jones et al. [23] presented that the BGM is well fit of the cow data. The parameter estimates from the model are illustrated in Table 2. The comparison among BIWM and other three models are given in Table 5.Figure 2
Cow data: marginal and joint contours, histogram fitted for bivariate mixture of inverse Weibull distribution. (a) Age: marginal contourh(x). (b) Location: marginal contour h(y). (c) Perspective joint contour. (d) Cow data. (e) Image joint contour.
(a)
(b)
(c)
(d)
(e)Table 3
One-sample Kolmogorov–Smirnov test, AIC, CAIC, BIC, and HQIC off(x).
K-S
P value
AIC
CAIC
BIC
HQIC
D = 0.16982
0.6842
104.5092
105.4323
106.0544
104.5883Table 4
One-sample Kolmogorov–Smirnov test, AIC, CAIC, BIC, and HQIC off(y).
K-S
P value
AIC
CAIC
BIC
HQIC
D = 0.19619
0.5075
141.6888
142.6119
143.234
14107679Figure 3
Empirical CDF, fitted IW CDF, density of histogram, and fitted IW density for the datax.
(a)
(b)Figure 4
PP-Plot fitted IW CDF, the QQ-Plot, and quantile of IW for the datax.
(a)
(b)Figure 5
Empirical CDF, fitted IW CDF, density of histogram, and fitted IW density for the datay.
(a)
(b)Figure 6
PP-Plot fitted IW CDF, the QQ-Plot, and quantile of IW for the datay.
(a)
(b)Table 5
The comparison among BIWM, BGM, MIW, and BIW based on AIC, BIC, CAIC, HQIC, KS, and PV for cow data.
Distributions
AIC
BIC
HQIC
CAIC
KS
PV
BIWM
351.1678
358.8937
351.5634
360.1678
0.963
0.78
BGM
355.8220
361.4812
356.298
362.4862
0.719
0.53
MIW
358.0725
360.6582
358.8159
361.9375
0.135
0.29
BIW
362.9531
368.5824
363.9281
369.7283
0.095
0.18


## 7. Conclusions

In this paper, we suggest BIWM distribution as a new finite mixture bivariate model. Some properties of the model have been derived such as the marginal probability density ofX or Y, cumulative distribution function, product moments about zero, the expected value of X or Y, the joint moment-generating function, the variance of X or Y, and the covariance between X and Y. The correlation coefficient has been studied for the two variables X and Y. It is noted that the correlation coefficient depends on the parameters. The EM algorithm estimation has been used to estimate the parameters of the BIWM. Cow data have been used to illustrate the importance and flexibility of the BIWM model. It is found that the BIWM model performs the best among the competitive distributions such as BGM, MIW, and BIW distributions.



---

*Source: 5234601-2020-03-28.xml*