Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2020 / Article

Research Article | Open Access

Volume 2020 |Article ID 8765321 | https://doi.org/10.1155/2020/8765321

Ammara Nawaz Cheema, Muhammad Aslam, Ibrahim M. Almanjahie, Ishfaq Ahmad, "Bayesian Modeling of 3-Component Mixture of Exponentiated Inverted Weibull Distribution under Noninformative Prior", Mathematical Problems in Engineering, vol. 2020, Article ID 8765321, 11 pages, 2020. https://doi.org/10.1155/2020/8765321

Bayesian Modeling of 3-Component Mixture of Exponentiated Inverted Weibull Distribution under Noninformative Prior

Academic Editor: Luis Payá
Received26 Mar 2020
Revised21 Jun 2020
Accepted10 Jul 2020
Published29 Jul 2020

Abstract

Bayesian study of 3-component mixture modeling of exponentiated inverted Weibull distribution under right type I censoring technique is conducted in this research work. The posterior distribution of the parameters is obtained assuming the noninformative (Jeffreys and uniform) priors. The different loss functions (squared error, quadratic, precautionary, and DeGroot loss function) are used to obtain the Bayes estimators and posterior risks. The performance of the Bayes estimators through posterior risks under the said loss functions is investigated through simulation process. Real data analysis of tensile strength of carbon fiber is also applied for 3 components to conclude the presentation of Bayes estimators. The limiting expressions are also elaborated for Bayes estimators and posterior risks in this study. The impact of some test termination times and sample sizes is reported on Bayes estimators.

1. Introduction

Mixture modeling exists in many situations, particularly whenever we have more than one subpopulation. Mixture densities have beautiful properties to solve the complex problems in an easier manner. Recently, [1] explored 3-component mixture modeling of exponentiated Weibull distribution under the Bayesian approach. Exponentiated inverted Weibull distribution (EIWD) has wide application in reliability theory. The authors in [2] used maximum likelihood and Bayes methods to derive parameters of EIWD under the type II censoring scheme. Later on, parameters of EIWD under type II censoring are discussed in [3]. The author in [4] derived Bayes and classical estimators of EIWD using noninformative prior. The authors in [5] proposed 2-parameter model of EIWD. Bayesian analysis of shape parameter of EIWD under different loss functions (LFs) is discussed in [6]. Bayes and E-Bayes estimators of EIWD using conjugate prior under different loss functions are estimated in [7]. The authors in [8] studied three-parameter weighted EIWD.

The number of components in a mixture distribution is due to heterogeneity of the parent population. It is often restricted to be finite, although in some cases the component may be infinite. As compared to simple modeling, it provides more attractive description of different statistical frameworks. Mixture modeling has wide applications in survival analysis. Recently, the authors in [9] studied 3-component mixture model of Pareto distribution by using type I right censoring scheme. Bayesian estimation and properties of 3-component mixture of Rayleigh distribution are discussed in [10]. The authors in [11] explored 3-component mixture of exponential distribution under different loss functions. Moreover, the authors in [12] performed Bayesian estimation for finite mixture of exponential, Rayleigh, and Burr type XII distribution.

Motivated by the abovementioned studies of 3-component mixture and EIWD, we investigate the Bayesian modeling of 3-component mixture of EIWD in this study. The main focus of this paper is to highlight efficient Bayes estimators (BEs) of component and proportional parameter(s). For this reason, two symmetric and two asymmetric LFs are used with noninformative priors, uniform prior (UP), and Jeffreys prior (JP), to obtain such results. The estimators are derived by applying the type I right censoring scheme.

The rest of the paper is designed as follows. In the next section, the 3-component mixture model of EIWD is designed. In Section 3, we illustrate the proposed BEs of different parameters under several LFs. The limiting expressions and simulation study are discussed in Sections 4 and 5. Real data analysis is presented in Section 6. In Section 7, conclusions are provided.

2. The 3-Component Mixture of the EIWD

For the shape parameter of a random variable , the pdf (probability density function) with the cdf (cumulative distribution function) of EIWD can be illustrated as

Here, for EIWD, is defined as shape parameter.

With and mixing proportion, a finite 3-component mixture model can be written as

For mixing proportion parameters and different component values, a 3-component mixture model of the EIWD is shown in Figure 1.

For 3-component mixture, the cdf is written as

3. Posterior Distribution Using the Noninformative Priors

The prior information plays an important role to differentiate between classical and Bayesian inference. The probability distribution which characterizes uncertainty of the parameter, prior to the existing information which is studied, is classified as prior distribution. A prior distribution is differentiated as noninformative prior, if it is smooth comparative towards likelihood function, whereas an informative prior (IP) is defined as a prior that has a contact towards the posterior distribution and is not the subject by the likelihood function. In this section, posterior distributions with the likelihood are obtained assuming the noninformative priors (NIPs) (UP and JP).

3.1. Likelihood Function

Assume that from the 3-component mixture modeling of EIWD n units are consumed in a life assessment process with fixed t (test termination time). Suppose that the selected trial reveals that n units from r failed till fixed t and the n-r where the rest of the units are still in running phase. It is noted that, due to the failures, from r failures, r1 failures classified are related to subpopulation I, r2 failures belong to subpopulation II, and r3 failures are related to subpopulation III. Now, the total uncensored sample points are classified as r=r1, r2, and r3, whereas the rest of the sample points n-r are considered as censored. Here, we have defined the time to failure, of the ith unit relating to lth subpopulation as xoi, 0 <Xoi ≤ t, where l = 1, …, 3 and i = 1, …, rl, and t is defined as time of test termination.

The likelihood of a 3-component mixture is stated as

By simplifying the expression, we obtain likelihood of 3-component mixture model of EIWD aswhereand for the uncensored observations are the failure times and .

3.2. Posterior Distribution Using Uniform Prior (UP)

The UP and JP are the most applied NIPs. Most of the researchers mentioned that UP is the most applied prior, for the evaluation, of required parameters of concern (see [1315]). The improper UP for the parametric component part , i.e., U (0, ∞), U (0, ∞), and U (0, ∞), is considered. The UP in the interval (0, 1) is applied for the parameters and , that is, U and U. The joint prior distribution of parameters can be stated as

So, under the UP, the joint posterior distribution of parameters can be written aswhereand is the beta function and extended as .

3.3. Posterior Distribution Using the Jeffreys Prior (JP)

The Jeffreys proposed a rule of thumb to specify noninformative prior for parameter θ as follows: If , then, , and if , then , using transformation of the variables (see [1619]). In respect of proportion parameters and , the JP is used as and . The joint prior distribution of parameters, with the assumption of independence of the said parameters , is stated as

Then, the joint posterior distribution becomeswhere

3.4. BEs and PRs under LFs

The real-valued function which illustrates a loss for estimator over the exact value of parameter is defined as loss function (LF). The current section discussed BEs and posterior risks (PRs) over four different LFs, that is, squared error loss function (SELF), quadratic loss function (QLF), precautionary loss function (PLF), and DeGroot loss function (DLF).

3.4.1. BEs and PRs among SELF

For d which is BE, PR can be written as , where is the SELF.

We obtain BEs and PRs assuming NIPs for the component and proportion parameters under SELF aswhere for the UP and for the JP.

3.4.2. BEs and PRs Assuming the UP and the JP under QLF

QLF is symmetric LF and mostly used in the least square theory. It needs more care to tackle due to its variance properties and being symmetric, where is the QLF. We can, respectively, define the BE and the PR under QLF as

By applying this concept using GP and ILP, the derivation of BEs and PRs iswhere and for the UP and JP.

3.4.3. BEs and PRs for the UP and the JP among PLF

PLF is an asymmetric LF, which firstly introduced by [20]. The general form of PLFs which is also a special case can be stated as . Under PLF, the derivation of BEs and PRs can be obtained with the help of

The derived results for BEs and PRs under the said priors and LF are observed aswhere for the UP and for the JP, respectively.

3.4.4. BEs and PRs Assuming the UP and the JP under DLF

Another asymmetric LF introduced by [21] is DLF, written as . The BEs and the PRs among DLF are obtained as

The derived BEs and PRs using the assumed priors arewhere .

4. Limiting Expressions

In regard to the uncensored sampling scheme, limiting expressions have wide applications. When test termination time it can be observed that uncensored points r approach n (sample size) and rl belongs to nl, where l = 1, …, 3. The points that are censored have turned out to be uncensored then and information given in the sample has also raised here. As a result, the effectiveness of the BEs is also raised due to the consideration of all the points in sample. Thus, limiting expressions for the NIPs can easily be obtained.

The limiting forms under UP and JP for the BEs and PRs are reported in Table 1.


ParametersBEsPRs
UPJPUPJP


5. Simulation Study

Simulation analysis is designed to check the presentation of BEs under UP and JP and with four different LFs for the 3-component mixture model of the EIWD:(i)At the first step, we have generated several sample sizes n = 50, 100, 200, and 500 assuming the different parametric values fixed as (, , , ) = {(4, 3, 2, 0.5, 0.3), (3, 3, 3, 0.4, 0.4), (2, 3, 4, 0.3, 0.5)}.(ii)By using Mathematica software, outcomes are averaged, by reiteration of simulation 1000 times.(iii)Randomly selected sample sizes belong to the first, second, and third factor densities, respectively.(iv)The right type I censoring procedure is applied to conclude the effect of t on BEs.(v)All observations which were greater than t have been considered as censored ones. Different fixed censoring times t are used to assess the effect of censoring rate on the estimates.(vi)In favor of the fixed t, values are taken as t = 25 and 30.(vii)The points those are larger than are taken as censored points.(viii)It is noted that only failures can be identified as a member of subpopulations I, II, and III of the 3-component mixture of EIWD.

In Table 2, results of simulation study are presented for .


tnLFsUPJP

2550SELFBEs3.4633.3622.9650.4674.3642.2681.9480.4570.3130.297
PRs0.6750.9430.9780.0050.9080.3670.3790.0050.0040.004
QLFBEs3.5262.7941.1820.4694.1492.0421.2550.4490.2660.285
PRs0.0410.0660.1110.0220.0520.0900.1250.0270.0610.047
PLFBEs3.6272.6012.4500.4843.7363.9191.4110.4930.2860.312
PRs0.1770.1760.2280.0110.1840.3340.1440.0110.0160.014
DLFBEs3.3752.7032.7240.4573.2282.7292.1910.4990.3120.333
PRs0.0450.0620.0830.0250.0430.0710.1000.0210.0460.042
100SELFBEs3.7562.8891.8840.4893.5102.4861.4610.4980.3180.302
PRs0.3130.3090.1970.0020.2870.1460.1420.0020.0020.002
QLFBEs4.0062.4741.5420.4683.5602.8292.3020.4880.2660.282
PRs0.0240.0400.0550.0120.0220.0410.0550.0110.0290.028
PLFBEs3.3942.7201.6070.4463.5742.7921.7770.4960.2960.337
PRs0.0830.0830.0770.0060.0810.1080.0940.0050.0070.007
DLFBEs4.4833.0241.5850.4742.8662.9052.9720.4630.3260.314
PRs0.0370.0370.0500.0130.0230.0330.0470.0120.0210.025
200SELFBEs3.5722.7581.7850.4943.2993.3831.6620.5170.2930.306
PRs0.1410.1380.0910.0010.1250.2330.0890.0010.0010.001
QLFBEs3.6182.3621.4680.4923.4622.9942.1980.4790.2990.261
PRs0.0110.0210.0250.0050.0110.0180.0290.0060.0120.016
PLFBEs3.4082.9872.0250.5033.8392.7311.7120.4720.3050.306
PRs0.0350.0510.0550.0020.0450.0500.0430.0020.0030.003
DLFBEs3.2413.1741.7150.4853.5542.7132.1620.4990.2840.311
PRs0.0110.0170.0250.0050.0110.0200.0250.0050.0140.012
500SELFBEs3.9722.4481.8530.5073.3652.9561.8270.4780.3030.292
PRs0.0530.0460.0390.0000.0550.0670.0350.0000.0000.000
QLFBEs4.1512.6861.7300.4813.9162.4622.2960.4830.3000.303
PRs0.0040.0070.0100.0020.0040.0070.0100.0020.0050.005
PLFBEs4.0392.3601.8520.4893.1672.7321.9100.4720.3080.300
PRs0.0160.0170.0190.0010.0150.0190.0190.0010.0010.001
DLFBEs3.0902.5271.8740.4903.2082.6972.1500.4750.3110.291
PRs0.0040.0070.0100.0020.0040.0070.0100.0020.0040.005

3050SELFBEs3.2581.9161.6480.4483.4542.1122.3350.4790.3000.326
PRs0.5310.2620.3020.0040.5180.3180.5450.0040.0040.004
QLFBEs4.7683.6681.2850.4884.1291.8961.2810.4360.2910.276
PRs0.0410.0710.1250.0210.0470.0710.1110.0250.0470.051
PLFBEs4.4792.8451.4810.4533.4243.1782.0800.4840.3060.333
PRs0.1970.1690.1260.0100.0140.2150.1930.0100.0130.013
DLFBEs4.7521.9411.2330.4972.4723.9222.2530.4900.3200.327
PRs0.0430.0660.1000.0220.0400.0620.0900.0190.0400.045
100SELFBEs3.4292.3681.7600.4604.6263.0952.4770.5040.2900.318
PRs0.2940.2070.1720.0020.4650.3680.3410.0020.0020.002
QLFBEs3.6272.5062.0940.4664.1082.4841.3800.4670.3000.293
PRs0.0220.0350.0520.0120.0230.0370.0580.0120.0240.025
PLFBEs3.5562.7601.9920.4913.1483.0672.7370.4910.3220.305
PRs0.0740.0920.0960.0050.0680.1030.1530.0050.0070.007
DLFBEs3.4362.8501.9870.4673.8364.7651.8360.4890.3190.319
PRs0.0220.0330.0450.0120.0200.0320.0500.0100.0210.022
200SELFBEs4.0252.6161.7050.4763.4192.3561.8240.4990.3110.300
PRs0.1930.1310.0760.0010.1310.1000.1000.0010.0010.001
QLFBEs3.3833.0072.2600.5054.0083.1372.2040.4830.2930.285
PRs0.0100.0180.0280.0050.0110.0180.0270.0050.0120.013
PLFBEs3.4783.1462.0660.0463.3042.5952.4840.4800.3120.317
PRs0.0420.0560.0530.0030.0400.0490.0690.0030.0030.003
DLFBEs3.2153.0032.1550.4623.5623.2972.1520.4770.3110.327
PRs0.0120.0170.0250.0060.0110.0180.0250.0060.0120.011
500SELFBEs3.2852.8721.8060.4803.9682.7792.1620.4870.3020.311
PRs0.0490.0590.0350.0000.0490.0570.0500.0000.0000.000
QLFBEs3.8342.7631.8230.4793.9172.8542.0870.4910.3000.300
PRs0.0040.0070.0100.0020.0040.0070.0110.0020.0040.005
PLFBEs3.6972.8742.1490.4733.7412.5891.8380.4810.3130.309
PRs0.0170.0210.0220.0010.0170.0180.0190.0010.0010.001
DLFBEs3.6012.6631.8580.4933.4242.8641.9900.4930.3020.294
PRs0.0040.0070.0100.0020.0040.0070.0100.0020.0050.005

From Table 2, it is examined that BEs assuming all stated NIPs and LFs are larger for the small n as compared to the highest n for t. It is noted that, for t, the variation of the BEs from supposed components is near to zero with the rise in n. However, the PRs assuming the mentioned priors and LFs decrease with the rise in n. In regard to the estimation of component part of the parameters, PRs give smaller results among the DLF, over to the results mentioned among the SELF, QLF, and PLF at different n and t. For proportion parameter estimation, it is reported that SELF illustrates minimum PRs compared to QLF, then PLF, and at last DLF. Therefore, results indicate that JP is the most suited prior over the UP for this study. On the ground of simulation formulation under the studied LFs, the DLF is considered the best for evaluating the component part of the parameters and SELF is observed to be efficient for proportion parameters.

6. Practical Application

EIWD has extensive uses in the area of tensile strength of carbon fiber. Therefore, in this view of analysis, we have used the data information of 100 sample points of tensile strength of carbon fiber. Earlier, this dataset is also studied by [22], which later has been considered by [5]. This dataset is based on the tensile strength of 100 observations of carbon fiber and is as follows: 3.7, 3.11, 4.42, 3.28, 3.75, 2.96, 3.39, 3.31, 3.15, 2.81, 1.41, 2.76, 3.19, 1.59, 2.17, 3.51, 1.84, 1.61, 1.57, 1.89, 2.74, 3.27, 2.41, 3.09, 2.43, 2.53, 2.81, 3.31, 2.35, 2.77, 2.68, 4.91, 1.57, 2.00, 1.17, 2.17, 0.39, 2.79, 1.08, 2.88, 2.73, 2.87, 3.19, 1.87, 2.95, 2.67, 4.20, 2.85, 2.55, 2.17, 2.97, 3.68, 0.81, 1.22, 5.08, 1.69, 3.68, 4.70, 2.03, 2.82, 2.50, 1.47, 3.22, 3.15, 2.97, 2.93, 3.33, 2.56, 2.59, 2.83, 1.36, 1.84, 5.56, 1.12, 2.48, 1.25, 2.48, 2.03, 1.61, 2.05, 3.60, 3.11, 1.69, 4.90, 3.39, 3.22, 2.55, 3.56, 2.38, 1.92, 0.98, 1.59, 1.73, 1.71, 1.18, 4.38, 0.85, 1.80, 2.12, and 3.65.

To demonstrate the proposed methodology, we have braked the given uncensored dataset in the 3-component parts according to right type I censoring scheme with rate of failure r = 91. It is unidentified which parametric part fails till a failure happens at or before t = 9. Having run the program in Mathematica software, the total tests are formulated 100 times:

Almost, here 9% censored observations are used. BEs and the PRs assuming the NIPs are reported in Table 3.


PriorLFs

UPSELFBEs8.535127.486146.793750.495140.30097
PRs1.428601.807822.197860.002440.00202
QLFBEs8.200417.003176.146720.485140.28312
PRs0.020000.033000.050000.010290.02376
PLFBEs8.618397.605976.953620.497570.30431
PRs0.166540.239570.319750.004840.00668
DLFBEs8.702477.727637.117260.500000.30769
PRs0.019230.031250.045450.009710.02184

JPSELFBEs8.367767.244666.470240.495140.30097
PRs1.400391.749502.093200.002400.00202
QLFBEs8.033056.761685.823210.485150.28713
PRs0.020410.034480.052630.010290.02376
PLFBEs8.451027.364416.630020.497570.30431
PRs0.166530.239510.319570.004840.00668
DLFBEs8.535127.486146.793750.500000.30769
PRs0.019600.032260.047620.009710.02184

From Table 3, it is noted that simulated results are compatible to real data study. There seem to be a few exceptions which are the reasons of the small set of data information. Both simulation and real data application results among the JP are the most accurate compared to the results under UP. However, DLF in contrast to other three LFs (SELF, QLF, and PLF) illustrates enhanced findings for the proportional part of parameters.

7. Conclusion

In this paper, the Bayesian formulation of the 3-component mixture model of EIWD under right type I censoring technique is studied. The comprehensive simulation process is built to evaluate and demonstrate several significant features about the BEs of the 3-component mixture model of EIWD assuming the NIPs under the different symmetric and asymmetric LFs (SELF, PLF, DLF, and QLF). Overestimation and underestimation of mixture proportion are inversely related to the sample size and are directly proportional to censoring rate. A small sample size and a large censoring rate cause the higher level of overestimation. But this effect can be reduced by using a large sample size. Posterior densities are derived and notified that they are in closed forms. The second aim of this paper was the selection of appropriate LF and prior for the inference of mixture parameters at different n and t. To judge the performance, we derived different posterior summaries, like BEs and their respective PRs by assuming different n and t. The limiting terms for the BEs and PRs of the shape parameters which are unknown here are also obtained among the said LFs (SELF, QLF, DLF, and PLF) assuming the NIPs (JP and UP). The compatible results are observed for simulation and real dataset analysis, to evaluate the performance of BEs. The contact of several n and t is estimated for BEs. The outcomes