Bayesian Analysis of Inverted KumaraswamyMixture Model with Application to Burning Velocity of Chemicals

Department of Mathematics & Statistics, International Islamic University, Islamabad 45320, Pakistan Department of Mathematics and Statistics, PMAS University of Arid Agriculture, Rawalpindi, Pakistan Department of Information Technology, Faculty of ICT, Baluchistan University of Information Technology Engineering and Management Sciences, Quetta, Pakistan College of Underwater Acoustics Engineering, Harbin Engineering University, Harbin, Heilongjiang, China Department of Computer Science, Faculty of ICT Baluchistan University of Information Technology Engineering and Management Sciences, Quetta, Pakistan


Introduction
Mixture models appear as obvious candidates whenever datasets that consist of two or more heterogeneous populations are mixed together. Due to its modeling versatility, the finite mixture model has attracted a great deal of attention in the history of statistics. To analyze the heterogeneous nature of processes, the mixture models are comparatively more suitable than the simple models. A mixture model with finite components is suitable to use when data are overdispersed, to fit a zero-expansion model, to measure heavy-tailed density, and to test for heterogeneity in cluster analysis. Mixture models have been effectively used in many areas such as industrial engineering (Ali et al. [1]), biology (Bhattacharya [2]), social sciences (Harris [3]), economics (Jedidi et al. [4]), and reliability (Sultan et al. [5]). For more detail about the finite mixture models, see Everitt [6], Ali [7], Feroze and Aslam [8], Zhang and Huang [9], Fundi et al. [10], Tripathi et al. [11], Noor et al. [12], and Feroze and Aslam [13].
Many researchers have provided valuable literature on inverted distribution, for example, Aljuaid [14] studied inverse Weibull, Noor and Aslam [15] analyzed inverse Weibull mixture distribution, Abd EL-Kader et al. [16] analyzed inverted Pareto type I distribution, Basheer [17] proposed generalized alpha power inverse Weibull distribution, and Hassan and Zaky [18] present study on estimation of entropy for inverse Weibull distribution under multiple censored data. Kumaraswamy [19] proposed a distribution which has widespread applications, particularly in situations that are bounded from below and above, such as individual's height, test scores, atmospheric temperature, and hydrological data. AL-Fattah et al. [20] obtained the IKum distribution from the Kumaraswamy distribution using the transformation X � (1/T) − 1 when random variable T has Kumaraswamy distribution with α and β as shape parameters. ey discussed important properties of inverted Kumaraswamy distribution and obtained parameters of the proposed model by using MLE and Bayesian technique. e IKum distribution has a long tail to the right; as a result, it can effectively be used for long-term reliability predictions and producing optimistic predictions as compared to other distributions.
Censoring is an important factor of experiments measuring life/failure times. Censored samples are encountered in life test whenever the experimenter has some obligations on the cost or available time for the experiment. Different censoring schemes are used for different experiments, but type I censoring is the most commonly used censoring scheme.
Our aim is to analyze inverted Kumaraswamy distribution in a different way as no other work after AL-Fattah et al. [20] is found on the IKum distribution. We propose a mixture model whose component densities are formed by IKum density and estimate the parameters and reliability function of the mixture model under study using Bayesian as well as frequentist method.

Two-Component Mixture Model of IKum Distribution.
A random variable X supposed to have a k component mixture model is defined as follows: where Θ � (δ i , α i , β i ), i � 1, 2, . . . , k e probability density function (pdf ) and reliability function of the mixture model whose component densities are characterized by IKum distribution are given by where pdf and reliability function of i th IKum density, respectively, are  [21], in many problems, only the futile (useless) items are easily marked as a family of first population and second population. For example, an engineer may divide failed items of electronic as a first population and second population on the basis of failure cause. From the whole population, s 1 units belong to the first population, s 2 are from the second population, and "m − s" items do not give us any information about the population to which they belong to. It is obvious that s � s 1 + s 2 are the number of uncensored items. Suppose that x ij denote the failure times of j th item which are belonging to the i th subpopulation and that x ij ≤ t 0 , i � 1, 2 and j � 1, . . ., s i . e likelihood function for the IKum mixture model using the above-discussed sampling scheme given by Mendenhall and Hader [21] is given by We use the SAS package to compute ML estimates of the parameter and their MSEs. where

Bayes Estimation.
e Bayesian approach is a powerful statistical tool used to reduce uncertainty in complex problems. Bayesian theory basically relies upon prior distribution and the use of loss functions. Loss function represents the loss incurred when the real parameter is derived from the estimated value. Square error loss function (SELF) used in the study is a symmetric loss function. In many situations, overestimation is more serious than underestimation, or vice versa. Asymmetric loss functions are those loss functions in which negative and positive errors of the same or different dimensions cause different losses. To compensate the situation, an asymmetric loss function is also used.

Posterior Density Assuming Informative (Gamma)
Prior. It is assumed that α i and β i each have gamma prior distribution with (a i , b i ) and (c i , d i ) hyperparameters, respectively, and δ 1 assumes a uniform prior so joint prior density for α 1 , α 2 , β 1 , β 2 , and δ 1 is us, posterior density using the likelihood function and joint prior in proportional form is as follows: Mathematical Problems in Engineering 3 Integration of the posterior density does not produce estimators in compact and simple form; therefore, we use Lindley's approximation to obtain Bayes estimators, posterior risks, and reliability estimates for the shape parameters of the IKum mixture model.

Lindley's Procedure for Estimation of Parameters.
Lindley [22] proposed an approximation known as Lindley's approximation used to conduct posterior analysis when posterior density involves a complex integral. In this approximation, Bayes estimator expands as a function that involves a posterior mode of Θ. Lindley's approximation has been utilized by many authors for the estimation of the parameters for the simple as well as mixture models; see Jaheen [23], Ahmad et al. [24], Sultan et al. [25], etc.
Consider the following integral where Θ � (α 1 , is an arbitrary function of Θ, and Q(Θ) is the logarithm of a posterior function for n observation. Lindley [22] suggested the following approximate Bayes estimator under SELF: where i, j, s � 1, 2, . . . , m All the functions on the right-hand side are to be obtained as the posterior mode. Q ijs is given in Appendix B. Parameters of the proposed IKum mixture model using Lindley's approximation may be obtained as where After equating U(Θ), U ik � 0, i, k � 1, 2, . . ., 5, and D � 0, so the Bayes estimators of parameters α 1 , α 2 , β 1 , β 2 , and δ 1 of the IKum mixture model under SELF are given by where B i , i � 1, 2, . . . , 5 are given earlier and τ ik are the elements of the inverse of the matrix Q ik . Posterior risk under SELF is the variance which can be evaluated as Under Linex loss function, equation (14) can be written as e Bayes estimators of IKum distribution under LLF are and posterior risks under LLF are Mathematical Problems in Engineering B i are defined earlier, τ ik are the elements of the inverse of the matrix Q ik , and D LLF and l i , k i , m i , z i , θ i where i � 1, 2 are given in Appendix B.

Posterior Density Assuming Informative (Inverse Beta)
Prior. It is assumed that the shape parameters α i and β i have inverse beta prior with hyperparameters a i , b i , and c i d i , respectively, Joint prior density for α i , β i , and δ i , is given by Combining the likelihood function (4) and joint prior (32), the following joint posterior density of the IKum mixture model is obtained: e posterior density in equation (33) again does not produce Bayes estimators in explicit form, so we use Lindley's approximation given in equation (14). e final form of Bayes estimators, posterior risks, and reliability function of IKum distribution under SELF and LLF assuming informative (inverse beta) prior is the same as given in (14)-(30).

Posterior Density Assuming Noninformative (Uniform)
Prior. Noninformation priors are an important part of the Bayesian tool and are considered for Bayesian analysis when there is little or no prior information available. Let Assuming independence combining the prior with likelihood function (4). e joint posterior density of α i , β i , and δ i is obtained as

Reliability Estimation.
e objective of assessing the reliability of estimates is to determine how much of the variability in the data is due to errors in measurement. And how much is in the true parameters. Approximate Bayes estimator of reliability function of IKum at some value t can be obtained as Here, where B i is defined earlier, τ ik are the elements of the invers of the matrix Q ik , and l i , Generate a random sample of different selected sample sizes from the proposed mixture model using the inverse transformation method If u 1 < δ 1 , then use u 1 to generate random variate x from the mixture of two-component IKum as

Results and Discussion
. If u 1 ≥ δ 1 , then use u 2 to generate random variate x from the mixture of two-component IKum as . Select a sample censored at a fixed test termination time t and only take censored observations. For the different choice of parameters, hyperparameter for the informative priors (gamma) are selected for i � 1, 2 to satisfy E(β i ) � c i /d i and inverse beta e above steps are repeated 1000 times. e Bayes estimates are computed over 1000 repetitions by averaging the estimate and the squared deviation, respectively. Estimates are computed using two informative (gamma and inverse beta) priors and uniform noninformative prior.
Results presented in Tables 1 and 2 (Appendix A) are obtained through simulation procedure which narrates the properties of the derived Bayes estimators and posterior risks of parameters and reliability estimate of the IKum mixture model. Different sample sizes, i.e., n � 30, 50, and 100 are taken to perform a simulation study. It is observed that as we increase the sample size, the estimate of parameters converges to a true parametric value. It is also observed that the use of LLF assuming gamma prior produces less posterior risk, hence can be thought of as a best loss function. An experimenter always tries to choose such a loss function for which he has to bear the minimum loss for estimation. In the same context, gamma prior resulted in smaller posterior risks as compared to other priors.
Bayes estimates are found overestimated for few cases and underestimated for few cases. It is all because the complex mixture model of IKum densities is considered and these Bayes estimates are obtained by using the approximate method. MLEs obtained in the simulation study are found to be somewhat inconsistent, and when compared with Bayes estimates, we found that PRs are lesser than MSEs of MLEs.
Results of simulation are also exhibited graphically which are given in Figure 1. From graphs of simulated Bayes estimates, we can see that we almost get the same results as obtained under simulation study numerically. It means we can obtain the same Bayes estimates through graphs.

Real Dataset Example.
is dataset consists of 56 observations related to the burning velocity of different chemical materials. e burning speed/velocity is the laminar flame speed under the specified composition, temperature, and pressure conditions. It decreases as the inhibitor concentration increases and can be checked by analyzing the pressure distribution in the spherical vessel and by observing the flame propagation directly. Data related to burning velocity (cm/sec) of different chemical materials are given in Table 3 and are available at http:// www.cheresources.com/mists.pdf. e following information is extracted from the above real data to analyze the mixture of the IKum model when T � 85: Mathematical Problems in Engineering And we obtain the following required numerical quantities when T � 70: Similarly considering T � 65, we obtain To analyze the data of the burning velocity of chemical materials, we consider three different censoring times. e data thus acquired are used to get MLEs and BEs of the parameter of the IKum mixture model that are given in Table 4. Bayes estimates are obtained assuming uniform, gamma, and inverse beta priors. However, two loss functions SELF and LLF are used for Bayesian estimation. It is found that MLEs are a bit lower than Bayes estimates. e mixture model comprises five parameters whose all four parameters are shape parameters except mixing weight. Mixing weight which is considered 0.40 in the mixture data is almost ideally estimated. Shape parameter α is estimated to be about on average 1.5 to 1.8 cm/sec for the first component, and for the second component density, it ranges from almost 0.90 to 1.00 cm/s.

Simulation Study of Real Data.
In this section, using the estimates of the real dataset, we determine the estimates through a simulation study. It is observed that the Bayes estimators obtained through simulation are very close to the true values of estimators. One can easily observe that the suitable prior for these data is gamma prior and the best loss function is LLF because they provide less posterior risk. Results are given in Table 5.     68  61  64  55  51  68  44  50  82  60  89  61  54  166  66  50  87  48  42  58  46  67  46  46  44  48  56  47  54  47  89  38  108  46  40  44  312  41  31  40  41  40  56  45  43  46  46  46  46  52  58  82  71  48 39 41

Conclusion
In this paper, we conduct a Bayesian estimation of the unknown parameters and reliability function of the inverted Kumaraswamy mixture model under type 1 right censoring. For the choice of different sample sizes, n � 30, 50, and 100 are taken to perform a simulation study. It is observed as the sample size increases the parameters converge to their true parametric value. And it is also noted that LLF is found to be the best loss function assuming gamma prior because it has a less posterior risk.
Bayes estimates are found overestimated for some values and underestimated for few values. From the real dataset, it is observed that as the censoring times T � 85, 70, and 65 decrease, the posterior risk also decreases respectively. e BEs α 1 , α 2 , β 1 , and β 2 represent the mean value of the burning velocity of the chemical material. And R(t) represents the reliability of the estimates. Simulation of real data is carried out to compare the parametric values. Graphical representation of parameters is also presented by taking the number of iterations on the x-axis and different parametric values on the y-axis. From these graphs, we Mathematical Problems in Engineering