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

Authors: A. S. Al-Moisheer; K. S. Sultan; M. A. Al-Shehri
Journal: Mathematical Problems in Engineering
 (2017)

---

## 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.


---

## Body

## 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: (1)fx;Θ=∑i=12pifix,Θi,0≤p1≤1,p1+p2=1,where the (pdf) of the first component (inverse Weibull) is given by(2)f1x;Θ1=β1α1-β1x-β1+1e-α1x-β1,x≥0,α1,β1>0,and the (pdf) of the second component (inverse Burr) is given by(3)f2x;Θ2=β2α2x-β2+11+x-β2-α2+1,x≥0,α2,β2>0,where Θ=(p1,α1,β1,α2,β2), Θ1=(α1,β1), and Θ2=(α2,β2). Evidently, the cumulative density function (cdf) from the MIWIBD is as follows: (4)Fx;Θ=∑i=12piFix,Θi,where the cdf for each distribution from the MIWIBD alone, respectively, is as follows: (5)F1x;Θ1=e-α1x-β1,x≥0,α1>0,β1>0,(6)F2x;Θ2=1+x-β2-α2,x≥0,β2>0,α2>0.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 Section2, 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: (7)EX=p1α1Γ1-1β1+p2α2Γα2-1β2Γ1+1β2,β1>1,β2,α2>1,VarX=p1α12Γ1-2β1-Γ21-1β1+p2α2Γα2-2β2Γ1+2β2-α22Γ2α2-1β2Γ21+1β2,β1>2,β2,α2>2,where Γ(·) denotes the gamma function.


### 2.2. Measures of Location (Mode and Median)

By solving the nonlinear equations with respect tox and from (4), the mode and median of the MIWIBD are obtained, respectively, by(8)p1β1α1-β1x-β1+2e-α1x-β1-β1+1+β1α1-β1x-β1-p2x-β2+21+x-β2-α2+1β2α2+1x-β21+x-β2-1-β2+1=0,(9)p1e-α1x-β1+p21+x-β2-α2=0.5.Table1 shows the modes and median of the MIWIBD for some selections of the parameters.Table 1
Numerical results for the mode(s) and median of the MIWIBD.
Θ
=
(
p
1
,
α
1
,
β
1
,
α
2
,
β
2
)
Mode(s)
Median
0.2,1
,
1.5,2
,
3
1.0517
1.8332
0.3,1
,
1.5,2
,
3
1.0695
1.7666
0.5,1
,
1.5,2
,
3
1.1257
1.6270
0.6,1
,
1.5,2
,
3
1.1717
1.5554
0.8,1
,
1.5,2
,
3
1.3660
1.4125
0.2,1
,
0.75,4
,
6
3.2311
,
8.5172
1.6876
0.3,1
,
0.75,4
,
6
3.2311
,
8.7123
1.6857
0.5,1
,
0.75,4
,
6
3.2311
,
9.0552
1.6804
0.6,1
,
0.75,4
,
6
3.2311
,
9.2414
1.6766
0.8,1
,
0.75,4
,
6
3.2311
,
9.7475
1.6637In Table1, the five parameters p1,α1,β1,α2, and β2 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 p1, but the median was affected by changing the value of p1. 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.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.
(a)
(b)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.
(a)
(b)


### 2.3. Reliability and Failure Rate Functions

The following equation gives the reliability function of the MIWIBD(10)Rx=p11-e-α1x-β1+p21-1+x-β2-α2,x≥0.Equations (1) and (4) help us in finding the failure rate function (hazard rate function HRF) of the MIWIBD and are given as (11)rx=p1β1α1-β1x-β1+1e-α1x-β1+p2β2α2x-β2+11+x-β2-α2+1p11-e-α1x-β1+p21-1+x-β2-α2,x≥0.The above equation is expressed by viewing the result by Al-Hussaini and Sultan [10], as (12)rx=hxr1x+1-hxr2x,where(13)hx=11+p2R2x/p1R1x,rix=fixRix,i=1,2,R1x=1-e-αx-β,R2x=1-1+x-β2-α2.By taking the derivative of the failure rate function, we get(14)r′x=hxr1′x+1-hxr2′x-hx1-hxr1x-r2x2.Observe thath(x) and 1-h(x) assume values in interval 0,1∀x. Also, it follows from (14) that if the derivative of the failure rate function is less than zero (ri′(x)<0,∀x, i=1,2), then the derivative of the failure rate function is less than zero (r′(x)<0,∀x). After few conversions, the derivative of the failure rate function can be reduced and is given in (14) where h(x) is determined in (12) and the derivative of the failure rate function ri′(x),i=1,2, is as follows:(15)r1′x=r1xx-β1+1β1α1-β11-e-α1x-β1-β1+1xβ1,r2′x=r2xx-β2+1β2α2+11+x-β2-1-α1-1β2+1xβ21-1+x-β2-α2.Equation (11) that represents the failure function of the MIWIBD holds for the following limits.Lemma 1.
We have(16)limx→0rx=0,if β2α2>1, where β2 and α2 are the shape parameters in IBD.
And(17)limx→∞rx=0.Proof.
First from (12) we see that limx→0h(x)=p1; then from the mixture components, IWD and IBD, respectively, we have limx→0⁡r1x=0 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 form(18)r2x=β2α2x-β2+11+x-β2-α2+11-1+x-β2-α2,and then in the IBD, we have three cases for the limit dependent on the multiple of two shape parameters β2, α2 and the limit for r2(x) can be written in the following form: (19)limx→0r2x=∞,β2α2<1,1,β2α2=1,0,β2α2>1,so we put the condition on the two shape parameters β2α2>1 in the (IBD) for limx→∞⁡r2(x)=0; thus (16) is proved. Second, from (12) it can be shown that limx→∞⁡h(x)≤1, which means 0≤hx≤1. Then for (IWD) we can see that limx→∞⁡r1(x)=0 [see Sultan et al. [1]]. Also for IBD the limit of failure rate function r2(x) is in (18). Simply by using Taylor expansion for (1+x-β2)α2+1 then limx→∞⁡r2(x)=0, so (17) is proved, and thus the proof is complete.


### 2.4. Performance of the Failure Rate Graphs

Ifxi∗,i=1,2, refers to the mode of the pdf fix, x1=min⁡(x1∗,x2∗) and x2=max⁡(x1∗,x2∗). Also, ri(x)=fi(x)/Ri(x); we observe both f1x and f2x in the numerator of rix↑ in 0,x1, with the denominator ↓ in the same interval. Finally, rx↑ is in 0,x1. Moreover, as x→∞, r(x)→0. In such case of the interval (x1,∞), two statuses appear.(a) Unimodal Status.Here x∗ defines the maximum point of the failure rate of the MIWIBD. In the interval (x1,x∗) the difference Δ between r1x and r2x is small so that the first two terms of the derivative of the failure rate function r′x in (14) dominate the third term and then the derivative of the failure rate function r′x>0. When the difference Δ increases to the point that the third term in the derivative of the failure rate function r′x dominates the first two terms, then the derivative of the failure rate function r′x<0 in (x∗,∞). Summarizing, we have the failure rate of the MIWIBD to be ↑ in (0,x∗) and ↓ in (x∗,∞), reaching zero as x→∞; see Figure 1(b).(b) Bimodal Status. Here x∗ and x∗∗ refer to, respectively, the smallest and largest maximum points of the failure rate of the mixture. When the difference Δ between r1x and r2x in the interval (x1,x∗) is small, where x1<x∗<x2<x∗∗, the third term in (14) is dominated by the first two terms and so r′x>0 in (0,x∗). The difference Δ in the interval (x∗,x∗∗∗), where x∗∗∗ is the local minimum point of rx, becomes larger to the point that the third term in r′x dominates the first two terms resulting in r′x<0 in (x∗,x∗∗∗). In the interval (x∗∗∗,x∗∗), the difference becomes small so that the third term in r′x is dominated by the first two terms, and so r′x>0. Summarizing, we have the failure rate of the mixed model to be ↑ in (0,x∗), decreasing in (x∗,x∗∗∗), ↑ in (x∗∗∗,x∗∗), and ↓ again in (x∗∗,∞), reaching 0 as x tends to ∞; see Figure 2(b).We observe, from Figures1(b) and 2(b), the shape of the model (unimodal and bimodal) is influenced by the parameters selected. Clearly, when (β1,α2,β2) 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.


## 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: (7)EX=p1α1Γ1-1β1+p2α2Γα2-1β2Γ1+1β2,β1>1,β2,α2>1,VarX=p1α12Γ1-2β1-Γ21-1β1+p2α2Γα2-2β2Γ1+2β2-α22Γ2α2-1β2Γ21+1β2,β1>2,β2,α2>2,where Γ(·) denotes the gamma function.


## 2.2. Measures of Location (Mode and Median)

By solving the nonlinear equations with respect tox and from (4), the mode and median of the MIWIBD are obtained, respectively, by(8)p1β1α1-β1x-β1+2e-α1x-β1-β1+1+β1α1-β1x-β1-p2x-β2+21+x-β2-α2+1β2α2+1x-β21+x-β2-1-β2+1=0,(9)p1e-α1x-β1+p21+x-β2-α2=0.5.Table1 shows the modes and median of the MIWIBD for some selections of the parameters.Table 1
Numerical results for the mode(s) and median of the MIWIBD.
Θ
=
(
p
1
,
α
1
,
β
1
,
α
2
,
β
2
)
Mode(s)
Median
0.2,1
,
1.5,2
,
3
1.0517
1.8332
0.3,1
,
1.5,2
,
3
1.0695
1.7666
0.5,1
,
1.5,2
,
3
1.1257
1.6270
0.6,1
,
1.5,2
,
3
1.1717
1.5554
0.8,1
,
1.5,2
,
3
1.3660
1.4125
0.2,1
,
0.75,4
,
6
3.2311
,
8.5172
1.6876
0.3,1
,
0.75,4
,
6
3.2311
,
8.7123
1.6857
0.5,1
,
0.75,4
,
6
3.2311
,
9.0552
1.6804
0.6,1
,
0.75,4
,
6
3.2311
,
9.2414
1.6766
0.8,1
,
0.75,4
,
6
3.2311
,
9.7475
1.6637In Table1, the five parameters p1,α1,β1,α2, and β2 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 p1, but the median was affected by changing the value of p1. 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.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.
(a)
(b)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.
(a)
(b)


## 2.3. Reliability and Failure Rate Functions

The following equation gives the reliability function of the MIWIBD(10)Rx=p11-e-α1x-β1+p21-1+x-β2-α2,x≥0.Equations (1) and (4) help us in finding the failure rate function (hazard rate function HRF) of the MIWIBD and are given as (11)rx=p1β1α1-β1x-β1+1e-α1x-β1+p2β2α2x-β2+11+x-β2-α2+1p11-e-α1x-β1+p21-1+x-β2-α2,x≥0.The above equation is expressed by viewing the result by Al-Hussaini and Sultan [10], as (12)rx=hxr1x+1-hxr2x,where(13)hx=11+p2R2x/p1R1x,rix=fixRix,i=1,2,R1x=1-e-αx-β,R2x=1-1+x-β2-α2.By taking the derivative of the failure rate function, we get(14)r′x=hxr1′x+1-hxr2′x-hx1-hxr1x-r2x2.Observe thath(x) and 1-h(x) assume values in interval 0,1∀x. Also, it follows from (14) that if the derivative of the failure rate function is less than zero (ri′(x)<0,∀x, i=1,2), then the derivative of the failure rate function is less than zero (r′(x)<0,∀x). After few conversions, the derivative of the failure rate function can be reduced and is given in (14) where h(x) is determined in (12) and the derivative of the failure rate function ri′(x),i=1,2, is as follows:(15)r1′x=r1xx-β1+1β1α1-β11-e-α1x-β1-β1+1xβ1,r2′x=r2xx-β2+1β2α2+11+x-β2-1-α1-1β2+1xβ21-1+x-β2-α2.Equation (11) that represents the failure function of the MIWIBD holds for the following limits.Lemma 1.
We have(16)limx→0rx=0,if β2α2>1, where β2 and α2 are the shape parameters in IBD.
And(17)limx→∞rx=0.Proof.
First from (12) we see that limx→0h(x)=p1; then from the mixture components, IWD and IBD, respectively, we have limx→0⁡r1x=0 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 form(18)r2x=β2α2x-β2+11+x-β2-α2+11-1+x-β2-α2,and then in the IBD, we have three cases for the limit dependent on the multiple of two shape parameters β2, α2 and the limit for r2(x) can be written in the following form: (19)limx→0r2x=∞,β2α2<1,1,β2α2=1,0,β2α2>1,so we put the condition on the two shape parameters β2α2>1 in the (IBD) for limx→∞⁡r2(x)=0; thus (16) is proved. Second, from (12) it can be shown that limx→∞⁡h(x)≤1, which means 0≤hx≤1. Then for (IWD) we can see that limx→∞⁡r1(x)=0 [see Sultan et al. [1]]. Also for IBD the limit of failure rate function r2(x) is in (18). Simply by using Taylor expansion for (1+x-β2)α2+1 then limx→∞⁡r2(x)=0, so (17) is proved, and thus the proof is complete.


## 2.4. Performance of the Failure Rate Graphs

Ifxi∗,i=1,2, refers to the mode of the pdf fix, x1=min⁡(x1∗,x2∗) and x2=max⁡(x1∗,x2∗). Also, ri(x)=fi(x)/Ri(x); we observe both f1x and f2x in the numerator of rix↑ in 0,x1, with the denominator ↓ in the same interval. Finally, rx↑ is in 0,x1. Moreover, as x→∞, r(x)→0. In such case of the interval (x1,∞), two statuses appear.(a) Unimodal Status.Here x∗ defines the maximum point of the failure rate of the MIWIBD. In the interval (x1,x∗) the difference Δ between r1x and r2x is small so that the first two terms of the derivative of the failure rate function r′x in (14) dominate the third term and then the derivative of the failure rate function r′x>0. When the difference Δ increases to the point that the third term in the derivative of the failure rate function r′x dominates the first two terms, then the derivative of the failure rate function r′x<0 in (x∗,∞). Summarizing, we have the failure rate of the MIWIBD to be ↑ in (0,x∗) and ↓ in (x∗,∞), reaching zero as x→∞; see Figure 1(b).(b) Bimodal Status. Here x∗ and x∗∗ refer to, respectively, the smallest and largest maximum points of the failure rate of the mixture. When the difference Δ between r1x and r2x in the interval (x1,x∗) is small, where x1<x∗<x2<x∗∗, the third term in (14) is dominated by the first two terms and so r′x>0 in (0,x∗). The difference Δ in the interval (x∗,x∗∗∗), where x∗∗∗ is the local minimum point of rx, becomes larger to the point that the third term in r′x dominates the first two terms resulting in r′x<0 in (x∗,x∗∗∗). In the interval (x∗∗∗,x∗∗), the difference becomes small so that the third term in r′x is dominated by the first two terms, and so r′x>0. Summarizing, we have the failure rate of the mixed model to be ↑ in (0,x∗), decreasing in (x∗,x∗∗∗), ↑ in (x∗∗∗,x∗∗), and ↓ again in (x∗∗,∞), reaching 0 as x tends to ∞; see Figure 2(b).We observe, from Figures1(b) and 2(b), the shape of the model (unimodal and bimodal) is influenced by the parameters selected. Clearly, when (β1,α2,β2) 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 vectorΘ of the mixture density. Equation (1) is found on a random sample of size n. The MLE Θ^ is determined as a result of the likelihood equations(20)∂LΘ∂θi=0,i=1,2,3,4,5,or given by(21)∂log⁡LΘ∂θi=0,i=1,2,3,4,5,where(22)LΘ=∏j=1nfxj;Θexplains the likelihood function formed under the assumption of identically independent distributions (iid) data x1,…,xn. The likelihood function based on the mixture density in (1) is obtained by(23)LΘ=∏j=1np1f1xj;Θ1+p2f2xj;Θ2,where Θ1=(α1,β1) and Θ2=(α2,β2).By taking the derivative of the log-likelihood functionL∗=log⁡L(Θ) with respect to the five parameters from the MIWIBD then, the derivatives from the first order of L∗ become(24)∂L∗∂p1=∑j=1nωxj;Θ=0,∂L∗∂α1=∑j=1np1ϕ1xj;Θη1xj;Θ=0,∂L∗∂β1=∑j=1np1ϕ2xj;Θη1xj;Θ=0,∂L∗∂β2=∑j=1np2ψ1xj;Θη2xj;Θ=0,∂L∗∂α2=∑j=1np2ψ2xj;Θη2xj;Θ=0,where ω(xj;Θ), ϕ1(xj;Θ), ϕ2(xj;Θ), η1(xj;Θ), η2(xj;Θ), ψ1(xj;Θ), and ψ2(xj;Θ) are given, respectively, by(25)ωxj;Θ=f1xj;Θ1-f2xj;Θ2fxj;Θ,ϕ1xj;Θ=-β1α1-1+β1α1-β1+1xj-β1,ϕ2xj;Θ=β1-1-log⁡α1-log⁡xj+α1xj-β1log⁡α1xj,η1xj;Θ=f1xj;Θ1fxj;Θ,η2xj;Θ=f2xj;Θ2fxj;Θ,ψ1xj;Θ=β2-1+α2+1xj-β21+xj-β2-1-1log⁡xj,ψ2xj;Θ=α2-1-log⁡1+xj-β2,and f(xj;Θ), f1(xj;Θ1), and f2(xj;Θ2) 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 forΘ=(p1,α1,β1,β2,α2). Hence, the likelihood function is given by (3) in Section 3. The parameters p1,α1,α2,β1, and β2 are independent random variables for the prior distribution of Θ=(p1,α1,β1,β2,α2) represented by g(p1,α1,β1,β2,α2) as follows:(26)gΘ∝1,0<p1<1,αi>0,βi>0.Thus, the joint posterior density of the vectorΘ is obtained by multiplying (22) and (26) as follows:(27)qΘ∣x_∝gΘ∏j=1np1f1xj;Θ1+p2f2xj;Θ2.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 U≡U(Θ), where Θ=θ1,θ1,…,θm′ and U≡U(Θ) is a function of Θ. For the unknown five parameters’ status, the approximation form reduces to the following:(28)U~BLΘ=UΘ+12A+∑i=15Bidi,where i=1,2,…,5,(29)A=∑i=15∑j=55UijΘτij,di=∑j=15UjΘτij,Bi=∑j=15τjjQjjiΘ+2∑j=25τijQijiΘ+Ei,Ei=τ23Q23i+τ24Q24i+τ25Q25i+τ34Q34i+τ35Q35i+τ45Q45i,where Q(Θ) is the logarithm of a posterior function for n observations, forming a random sample X_, from a density f· for i,j,s=1,2,…,5,(30)UiΘ^=∂UΘ∂θiΘ=Θ^,UijΘ^=∂2UΘ∂θi∂θjΘ=Θ^,QijΘ^=∂2QΘ∂θi∂θjΘ=Θ^,QijsΘ^=∂3QΘ∂θi∂θj∂θsΘ=Θ^,Σm×m=τij=-Qijm×m-1.All terms on (28) are to be computed at the posterior mode since the logarithm of the posterior density in (27) is defined by(31)QΘ∣x_≡log⁡q∝∑j=1nlog⁡p1f1xj;Θ1+p2f2xj;Θ2.The mode of the posterior density can be obtained by solving the five nonlinear equationsi=1,2, the same as that mentioned before in Section 3 in (24) since the noninformative previous ones g(Θ)∝1, 0<p1<1, αi>0, and βi>0.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 (Qij)5×5,i,j=1,2,…,5, in (30). These elements are evaluated by (A.1) in Appendix A. Numerically by inverting the matrix -Qij, the elements τij,i,j=1,2,…,5, of the matrix Σ are computed where the elements Qijs,i,j,s=1,2,…,5, 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 p1,α1,α2,β1, and β2 is evaluated by equating U(Θ) in (28) to one of the five parameters, so that Uij=0 and A=0,i,j=1,2,…,5. (32)p1~=p^1+12∑i=15Biτi1,α1~=α^1+12∑i=15Biτi2,α2~=α^2+12∑i=15Biτi3,β1~=β^1+12∑i=15Biτi4,β2~=β^2+12∑i=15Biτi5,where Bi, i=1,…,5, are obtained in (29) and τij, i,j=1,2,…,5, are the elements of the inverse matrix Qij. In (31), the functions are calculated at the posterior mode.


## 5. 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.Table2 obtained the descriptive statistics for the carbon monoxide data.Table 2
Descriptive statistics for the carbon monoxide data of the MIWIBD.
Mean data
N
Mean
Mode
Median
Variance
St dev
Minimum
Maximum
Carbon monoxide level
151
2.0911
1.51
1.88
1.34
1.16
0.10
4.78The maximum likelihood estimates (MLEs) and Bayes estimates (Bs) for the MIWIBD are calculated in Table3.Table 3
MLEs and Bs for the carbon monoxide data of the MIWIBD.
Parameter
p
^
1
α
^
1
α
^
2
β
^
1
β
^
2
MLE
0.6995
0.5038
0.6461
2.7637
2.5551
Bayes estimate
0.6813
0.5058
0.7814
2.7275
2.6999From Figure3, 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.Table 4
MLEs and Bs (K-S) test for the carbon monoxide data of the MIWIBD.
(K-S) MLE
0.0978
(K-S) Bayes estimate
0.0995Figure 3
Empirical cdf for the fitted carbon data of the MIWIB model.From Table4, 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 matrixI(Θ) is used to determine the approximate100(1-δ) confidence intervals (CIs) of the parameters Θ as Θ^±ξδ/2V(Θ^), where V(Θ^) are the variances of the parameters given from I-1(Θ^) and ξδ/2 is the upper ξδ/2 percentile of the standard normal distribution. The variance-covariance matrix of Θ is calculated as(33)I-1Θ^=0.0065710.000881-0.0050740.052351-0.0189850.0008810.000697-0.0006780.003241-0.002062-0.005074-0.0006780.057469-0.0256780.0131210.0523510.003241-0.0256780.857329-0.255263-0.018985-0.0020620.013121-0.2552630.090989.The 90% and 95% CIs for the MLEs of the parameters are evaluated in Table5.Table 5
MLEs and Bs 90% CIs for the carbon monoxide data of the MIWIBD.
Parameter
p
^
1
α
^
1
α
^
2
β
^
1
β
^
2
90% CIs
0.5665,0.8324
0.0001,0.5468
0.1516,1.1405
2.3705,3.1568
1.0366,4.0735
95% CIs
0.5406,0.8583
0.4523,0.5552
0.0551,1.2370
2.2938,3.2335
0.7403,4.3698


## 6. 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 ofΘ and the performance of the failure rate functions of the 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.



---

*Source: 7824323-2017-10-31.xml*