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 variables x 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.
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.,
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 Section 2. 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:where θ > 0 and λ > 0 are the shape and scale parameters.
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 are independent.
The probability density function of bivariate inverse Weibull mixture distribution is given aswhere 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 bywith 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:
Let h(x, y) be the joint probability density function of (X, Y), then
Suppose that
Substituting a, b, c, and d in equation (7), we obtain
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 as b = 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.
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The marginal densities of X and Y, respectively, arewhere π1 = a + b, andwhere π2 = a + c.
The cumulative distribution function is given by
The survival function of BIWM distribution is given by
So,
The hazard rate function (hrf) is given as follows:
The conditional probability function of x for given y is given byand the conditional probability function of y for given x is given by
3. Moments
In this section, we derive the statistical properties of BIWM distribution. The product moments about zero is given by
Let z = λ1x−θ, = β1y−φ, and dz = −θλ1x−θ−1 dx, at x = 0, z = 0, x = ∞, z = ∞, y = 0, = 0, y = ∞, and = ∞; hence, we obtainand then
The expected values of X and Y are
The joint moment-generating function of bivariate mixture is given by
By using exponential expansion,
Then, we haveand then
Finally, we get
4. Correlation
The covariance between X and Y for the BIWM model is given by
So,
It is noted that the covariance between two variables X 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:where σX and σY are the standard deviation for X and Y, respectively. Then, the variance of X and Y are shown as follows:where
Then, we can obtain the standard deviation for X and Y by taking the positive squared root of and , respectively, as follows:
We compute numerically the correlation coefficient . The numerical results are listed in Table 1 for different values of the parameters.
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
The mixing proportions are , and .
We define ; then, the EM algorithm as a method of estimation is given by finding the complete log-likelihood ℓ as follows:
These group membership variables (ak, bk, ck) so in the E step have a linear relation, so are calculated asand 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
The M step is completed by setting
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 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.
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 , 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 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.
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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 of X 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.
Data Availability
The data used in the example application are given in the paper link: https://doi.org/10.1080/03610920008832636.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This project was supported by Jouf University, Deanship of Scientific Research, under the research project (40/327).