A Mixture of Inverse Weibull and Inverse Burr Distributions : Properties , Estimation , and Fitting

The newmixture model of the two components of the inverseWeibull and inverse Burr distributions (MIWIBD) is proposed. First, the properties of the investigated mixture model are introduced and the behaviors of the probability density functions and hazard rate functions are displayed.Then, the estimates of the five-dimensional vector of parameters by using the classical method such as themaximum likelihood estimation (MLEs) and the approximationmethod by using Lindley’s approximation are obtained. Finally, a real data set for the proposed mixture model is applied to illustrate the proposed mixture model.


Introduction
The importance of mixture models comes from the fact that most available data can be considered as data coming from a mixture of two or more statistical models; see Sultan et al. [1].For books that dealt with the models of the mixture, see Everitt and Hand [2] and McLachlan and Peel [3].Because the mixing of statistical distributions gives a new distribution with the properties of its compounds, we in this paper propose the two-component mixture models of inverse Weibull and inverse Burr distributions (MIWIBD).For the importance of the inverse Weibull distribution (IWD) as a single component from its uses in physical phenomena, see Keller et al. [4].Also, for the importance of the inverse Burr distribution (IBD) as one component from its uses in forestry applications, see Lindsay [5].This importance for each distribution alone has made us merge the two distributions together to obtain new properties from the distributive compounds.It should be noted that the mixing of the IWIBD gives a mixture model with a unimodal and bimodal peak for the hazard rate functions and these forms are important in applications which will be displayed in Section 2. The probability density function (pdf) from the MIWIBD is as follows: where the (pdf) of the first component (inverse Weibull) is given by and the (pdf) of the second component (inverse Burr) is given by where Θ = ( 1 ,  1 ,  1 ,  2 ,  2 ), Θ 1 = ( 1 ,  1 ), and Θ 2 = ( 2 ,  2 ).Evidently, the cumulative density function (cdf) from the MIWIBD is as follows: where the cdf for each distribution from the MIWIBD alone, respectively, is as follows: 1 (; Θ 1 ) =  −( 1 ) − 1 ,  ≥ 0,  1 > 0,  1 > 0, 2 (; Θ 2 ) = (1 +  − 2 ) − 2 ,  ≥ 0,  2 > 0,  2 > 0. (6) Some papers have dealt with the mixtures of two inverse Weibull distributions (MTIWD), for example, Sultan et al. [1] and Sultan and Al-Moisheer [6].In addition, there are some researches that have discussed the mixtures of two inverse Burr distributions (MTIBD), for example, the works of Al-Moisheer [7].Also, there is a mixture of one of its components which is IWD; see Sultan and Al-Moisheer [8].
In this paper, the order is as follows: in Section 2, we introduce few properties of the MIWIBD.In Section 3, through the method of maximum likelihood we find the five unknown parameters estimates of the MIWIBD.In Section 4, we use Lindley's approximation to estimate the unknown parameters of the MIWIBD.In Section 5, we apply the MIWIBD by fitting it to a real data collected from Jeddah city for measuring the carbon monoxide level in different locations.Finally, we draw expressions for Lindley's approximation matrix, and these are displayed in Appendix.

Some Properties for the MIWIBD
From (2) and (3), Keller et al. [4] and Abd-Elfattah and Alharbey [9] have discussed some properties of the IWD and IBD, respectively.In this section, we discuss some properties of the MIWIBD by merging the corresponding conclusions of the IWD and IBD.

Measures of Location and Dispersion (Mean and Variance).
The measures of location and dispersion for the mean and variance of the MIWIBD in (1) are as follows: where Γ(⋅) denotes the gamma function.

Measures of Location (Mode and Median)
. By solving the nonlinear equations with respect to  and from (4), the mode and median of the MIWIBD are obtained, respectively, by Table 1 shows the modes and median of the MIWIBD for some selections of the parameters.
In Table 1, the five parameters  1 ,  1 ,  1 ,  2 , and  2 are selected to display the unimodal and bimodal shapes for the pdf of the proposed MIWIBD.Table 1 clearly shows that the modes are not much affected by changing the value of  1 , but the median was affected by changing the value of  1 .Figures 1(a) and 2(a) show the pdf between the components and their mixtures with parameters displaying the shapes for the peak of the unimodal and bimodal cases for the proposed MIWIBD.

Reliability and Failure Rate Functions. The following equation gives the reliability function of the MIWIBD
Equations ( 1) and (4) help us in finding the failure rate function (hazard rate function HRF) of the MIWIBD and are given as The above equation is expressed by viewing the result by Al-Hussaini and Sultan [10], as where By taking the derivative of the failure rate function, we get Observe that ℎ() and 1 − ℎ() assume values in interval [0, 1] ∀.Also, it follows from (14) that if the derivative of the failure rate function is less than zero (   () < 0, ∀,  = 1, 2), then the derivative of the failure rate function is less than zero (  () < 0, ∀).After few conversions, the derivative of the failure rate function can be reduced and is given in (14) where ℎ() is determined in (12) and the derivative of the failure rate function    (),  = 1, 2, is as follows: Equation ( 11) that represents the failure function of the MIWIBD holds for the following limits.

Performance of the
(a) Unimodal Status.Here  * defines the maximum point of the failure rate of the MIWIBD.In the interval ( 1 ,  * ) the difference Δ between  1 () and  2 () is small so that the first two terms of the derivative of the failure rate function   () in ( 14) dominate the third term and then the derivative of the failure rate function   () > 0. When the difference Δ increases to the point that the third term in the derivative of the failure rate function   () dominates the first two terms, then the derivative of the failure rate function   () < 0 in ( * , ∞).Summarizing, we have the failure rate of the MIWIBD to be ↑ in (0,  * ) and ↓ in ( * , ∞), reaching zero as  → ∞; see Figure 1(b).
We observe, from Figures 1(b) and 2(b), the shape of the model (unimodal and bimodal) is influenced by the parameters selected.Clearly, when ( 1 ,  2 ,  2 ) varied from 1.5, 2.0, and 3.0 to 0.75, 4.0, and 6.0 the model is varied from the unimodal case to the bimodal case.

Classical Method for Estimating the Parameters from the MIWIBD
Here, we define the classical method estimation of the maximum likelihood approach for the five-dimensional parameter vector Θ of the mixture density.Equation ( 1) is found on a random sample of size .The MLE Θ is determined as a result of the likelihood equations or given by where explains the likelihood function formed under the assumption of identically independent distributions (iid) data  1 , . . .,   .The likelihood function based on the mixture density in ( 1) is obtained by where By taking the derivative of the log-likelihood function  * = log (Θ) with respect to the five parameters from the MIWIBD then, the derivatives from the first order of  * become where (  ; Θ),  1 (  ; Θ),  2 (  ; Θ),  1 (  ; Θ),  2 (  ; Θ),  1 (  ; Θ), and  2 (  ; Θ) are given, respectively, by and (  ; Θ),  1 (  ; Θ 1 ), and  2 (  ; Θ 2 ) are as in ( 1)-( 3), respectively.We can obtain the solutions for (24) to get the estimates of the five parameters from the MIWIBD and to solve them using Newton-Raphson method.

Bayesian Method by Using Lindley's Approximation for Estimating the Parameters from the MIWIBD
In Bayesian estimation the posterior distribution function is defined by multiplying the likelihood function with a prior distribution for Θ = ( 1 , Thus, the joint posterior density of the vector Θ is obtained by multiplying ( 22) and (26) as follows: From ( 27), we observe that the posterior density of the vector Θ is proportional to the likelihood function mentioned in Section 3.
Lindley's [11] approximation under the squared error loss function is evaluated to get the Bayes estimator of  ≡ (Θ), where Θ = ( 1 ,  1 , . . .,   )  and  ≡ (Θ) is a function of Θ.For the unknown five parameters' status, the approximation form reduces to the following: where  = 1, 2, . . ., 5,  = All terms on (28) are to be computed at the posterior mode since the logarithm of the posterior density in ( 27) is defined by The mode of the posterior density can be obtained by solving the five nonlinear equations  = 1,2, the same as that mentioned before in Section 3 in (24) since the noninformative previous ones (Θ) ∝ 1, 0 <  1 < 1,   > 0, and   > 0.

Application
In this section, we apply the real data collected to fit the proposed mixture model.We use the data collected from Jeddah city for measuring the carbon monoxide level in different locations during the period of January-June 2009 with sample size 151.Table 2 obtained the descriptive statistics for the carbon monoxide data.
The maximum likelihood estimates (MLEs) and Bayes estimates (Bs) for the MIWIBD are calculated in Table 3.
From Figure 3, the carbon monoxide data provides a suitable fit for the proposed mixture model under MLEs and Bayes estimates.In addition, we used Kolmogorov-Smirnov test (K-S) to fit the carbon monoxide data as shown in Table 4.
From Table 4, we observe that the values of the (K-S) test under the MLEs and Bayesian estimates give appropriate fit from the MIWIBD at 5% level of significance.
The Fisher information matrix (Θ) is used to determine the approximate100(1 − ) confidence intervals (CIs) of the parameters Θ as Θ± /2 √ ( Θ), where ( Θ) are the variances of the parameters given from  −1 ( Θ) and  /2 is the upper  /2 percentile of the standard normal distribution.The variancecovariance matrix of Θ is calculated as ) . The 90% and 95% CIs for the MLEs of the parameters are evaluated in Table 5.

Conclusion
In this paper, the MIWIBD are proposed and some important measures of the MIWIBD are discussed such as measures of locations and measures of dispersion.Also, numerical results for the mode and median of the MIWIBD are computed based on different choices of Θ and the performance of the failure rate functions of MIWIBD is interpreted through the plots.In addition, the estimates of the vector of the unknown parameters of the MIWIBD are given.Further, the MIWIBD are fitted to the data from Jeddah city for measuring the carbon monoxide level in different locations.Finally, the expressions for Lindley's approximation matrix are shown in Appendix.

Table 2 :
Descriptive statistics for the carbon monoxide data of the MIWIBD.

Table 3 :
MLEs and Bs for the carbon monoxide data of the MIWIBD.

Table 4 :
MLEs and Bs (K-S) test for the carbon monoxide data of the MIWIBD.