Abstract

The new mixture model of the two components of the inverse Weibull 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 the maximum likelihood estimation (MLEs) and the approximation method 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.

1. 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 byand the (pdf) of the second component (inverse Burr) is given bywhere , , and . 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:

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.

2. 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.

2.1. 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 denotes the gamma function.

2.2. 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 , and 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 , but the median was affected by changing the value of . 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.

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Figure 1 

(a) Density functions: components and their mixture with parameters (0.5, 1.0, 1.5, 2.0, and 3.0). (b) HR functions: components and their mixture with parameters (0.5, 1.0, 1.5, 2.0, and 3.0), unimodal case.

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Figure 2 

(a) Density functions: components and their mixture with parameters (0.5, 1.0, 0.75, 4.0, and 6.0). (b) HR functions: components and their mixture with parameters (0.5, 1.0, 0.75, 4.0, and 6.0), bimodal case.

2.3. 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 and assume values in interval . Also, it follows from (14) that if the derivative of the failure rate function is less than zero (,, ), then the derivative of the failure rate function is less than zero (,). After few conversions, the derivative of the failure rate function can be reduced and is given in (14) where is determined in (12) and the derivative of the failure rate function , is as follows:

Equation (11) that represents the failure function of the MIWIBD holds for the following limits.

Lemma 1. We have if , where and are the shape parameters in IBD.
And

Proof. First from (12) we see that ; then from the mixture components, IWD and IBD, respectively, we have for the first component (IWD) from the mixture [see Sultan et al. [1]]. Now, from (13) it can be shown the failure rate function for the second component (IBD) from the mixture takes a formand then in the IBD, we have three cases for the limit dependent on the multiple of two shape parameters , and the limit for can be written in the following form: so we put the condition on the two shape parameters in the (IBD) for ; thus (16) is proved. Second, from (12) it can be shown that , which means . Then for (IWD) we can see that [see Sultan et al. [1]]. Also for IBD the limit of failure rate function is in (18). Simply by using Taylor expansion for then , so (17) is proved, and thus the proof is complete.

2.4. Performance of the Failure Rate Graphs

If , refers to the mode of the pdf , and . Also, ; we observe both and in the numerator of in , with the denominator in the same interval. Finally, is in . Moreover, as , . In such case of the interval , two statuses appear.

(a) Unimodal Status. Here defines the maximum point of the failure rate of the MIWIBD. In the interval the difference between and 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 . When the difference 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 in . Summarizing, we have the failure rate of the MIWIBD to be in and in , reaching zero as ; see Figure 1(b).

(b) Bimodal Status. Here and refer to, respectively, the smallest and largest maximum points of the failure rate of the mixture. When the difference between and in the interval is small, where , the third term in (14) is dominated by the first two terms and so in . The difference in the interval , where is the local minimum point of , becomes larger to the point that the third term in dominates the first two terms resulting in in . In the interval , the difference becomes small so that the third term in is dominated by the first two terms, and so . Summarizing, we have the failure rate of the mixed model to be in , decreasing in , in , and again in , reaching as tends to ; see Figure 2(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 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.

3. 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 of the mixture density. Equation (1) is found on a random sample of size . The MLE is determined as a result of the likelihood equationsor given bywhereexplains the likelihood function formed under the assumption of identically independent distributions (iid) data . The likelihood function based on the mixture density in (1) is obtained bywhere and .

By taking the derivative of the log-likelihood function with respect to the five parameters from the MIWIBD then, the derivatives from the first order of becomewhere , , , , , , and are given, respectively, byand , , and 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.

4. 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 . Hence, the likelihood function is given by (3) in Section 3. The parameters , and are independent random variables for the prior distribution of represented by as follows:

Thus, the joint posterior density of the vector is obtained by multiplying (22) and (26) as follows:

From (27), we observe that the posterior density of the vector 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 , where and is a function of . For the unknown five parameters’ status, the approximation form reduces to the following:where ,where is the logarithm of a posterior function for observations, forming a random sample , from a density for ,

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 , the same as that mentioned before in Section 3 in (24) since the noninformative previous ones , , , and

To evaluate Lindley’s [11] approximation for the Bayes estimator of the vector of parameters form in (28), we define the elements of the matrix , in (30). These elements are evaluated by (A.1) in Appendix A. Numerically by inverting the matrix , the elements , of the matrix are computed where the elements , are defined by (B.1) in Appendix B.

Now, for the MTBIIID, Lindley’s [11] approximation for the Bayes estimator of the vector of five parameters of , and is evaluated by equating in (28) to one of the five parameters, so that and . where , , are obtained in (29) and , , are the elements of the inverse matrix . In (31), the functions are calculated at the posterior mode.

5. Application