Bayesian Prediction Based on Generalized Order Statistics from a Mixture of Two Exponentiated Weibull Distribution Via MCMC Sumulation

This paper is concerned with the problem of obtaining the maximum likelihood prediction (point and interval) and Bayesian prediction (point and interval) for a future observation from mixture of two exponentiated Weibull (MTEW) distributions based on generalized order statistics (GOS). We consider one-sample and two-sample prediction schemes using the Markov chain Monte Carlo (MCMC) algorithm. The conjugate prior is used to carry out the Bayesian analysis. The results are specialized to upper record values.


Introduction
In life testing, reliability and quality control problems, mixed failure populations are sometimes encountered.Mixture distributions comprise a finite or infinite number of components, possibly of different distributional types, that can describe different features of data.In recent years, the finite mixture of life distributions have to be of considerable interest in terms of their practical applications in a variety of disciplines such as physics, biology, geology, medicine, engineering and economics, among others.Some of the most important references that discussed different types of mixtures of distributions are Everitt and Hand (1981), Titterington et al. (1985), McLachlan and Basford (1988), Jaheen (2005), Jaheen and Mohammed (2011) and AL-Hussaini and Hussein (2012).Mudholkar and Sirvastava (1993) introduced a new distribution, called exponentiated Weibull (EW) distribution as an extension of the Weibull family of distributions by adding an additional shape parameter.This distribution is very important distribution in lifetime tests.The probability density function (PDF), cumulative distribution function (CDF) and reliability function (RF) are given below f (t) = αθt α−1 e −t α η θ−1 , t > 0, (α > 0), (θ > 0), ( 1) where α and θ are the shape parameters of the model.Also, the hazard rate function (HRF) takes the form where η = 1 − e −t α and H(•) = f (•) R(•) .The EW distribution includes a number of distributions as particular cases: if the shape parameter θ = 1, then the PDF is that Weibull distribution, when α = 1 then the PDF is that Exponentiated Exponential distribution, if α = 1 and θ = 1 then the PDF is that Exponential distribution and if α = 2 then the PDF is that One parameters Burr-X distribution.Mudholkar and Hutson (1996) showed that the density function of the EW distribution is decreasing when αθ ≤ 1 and unimodal when αθ > 1.Some statistical properties of this distribution are discussed by Jiang and Murthy (1999) and Nassar and Eissa (2003).Nassar and Eissa (2004) obtained Bayes estimators for the two shape parameters, the reliability, and the failure rate function of the EW distribution based on type-II censored and complete samples.Choudhury (2005) proposed a simple derivation for the moments of the EW distribution.Singh et al. (2005a, b) derived the maximum likelihood (ML) and Bayes estimates of the two and three parameters of the EW distribution based on type-II censored samples.Pal et al. (2006) introduced many properties and obtained some inferences for the three-parameter EW distribution.Kim et al. (2009) obtained the ML and Bayes estimators for the two shape parameters and reliability function of the EW distribution based on progressive type-II censored sample.
The property of identifiability is an important consideration on estimating the parameters in a mixture of distributions.A mixture is identifiable if there exists a one-to-one correspondence between the mixing distribution and a resulting mixture.That is, there is a unique characterization of the mixture.Therefore, a mixture of Exponentiated Weibull components is identifiable.Identifiability of mixtures has been discussed by several authors, including Teicher (1963), AL-Hussaini and Ahmad (1981) and Ahmad (1988).
Recently, there has been a considerable amount of interest in the Bayesian approach which allows prior subjective knowledge on lifetime parameters, as well as experimental data, to be incorporated into the inferential procedure.
For the Bayesian approach, a loss function must be specified.
A wide variety of loss functions have been developed in the literature to describe various types of loss structures.The balanced loss function was introduced by Zellner (1994).Jozani et al. (2006) introduced an extended class of the balanced loss function of the form where Υ(•) is a suitable positive weight function and Φ(Ψ(θ), δ) is an arbitrary loss function when estimating Ψ(θ) by δ.The parameter δ o is a chosen prior estimator of Ψ(θ), obtained for instance from the criterion of ML, least squares or unbiasedness among others.They give a general Bayesian connection between the case of Ω > 0 and Ω = 0 where 0 ≤ Ω < 1.
The purpose of this paper is to obtain the maximum likelihood and the Bayes prediction point and interval in the case of one-sample scheme and two-sample scheme.The point predictors are obtained based on balanced square error loss (BSEL) function and the balanced LINEX (BLINEX) loss function.We used ML to estimate the parameters, α and θ of the MTEW distribution based on GOS.The conjugate prior is assumed to carry out the Bayesian analysis.The results are specialized to record values.

Maximum Likelihood Estimation
Assuming that the parameters θ j and α j are unknown and p j is known, the likelihood equations are given, for j = 1, 2, by where, ≡ ln L(θ | t) and for j = 1, 2 Equations ( 12) do not yield explicit solutions for θ j and α j , j = 1, 2, and have to be solved numerically to obtain the ML estimates of the four parameters, Newton-Raphson iteration is employed to solve (12). Remark: The parameters of the components are assumed to be distinct, so that the mixture is identifiable.For the concept of identifiability of finite mixtures and examples, see Everitt and Hand (1981), AL-Hussaini and Ahmad (1981) and Ahmad and AL-Hussaini (1982).
This section deals with studying the maximum likelihood and the Bayes prediction point and interval in the case of one-sample scheme and two-sample scheme.

Prediction in Case of One-sample Scheme
Based on the informative T 1;n, m,k , T 2;n, m,k , ..., T r;n, m,k , GOS's from the MTEW distribution with two parameters, for the remaining (n − r) components, let T s;n, m,k , s = r + 1, r + 2, ..., n denote the future lifetime of the s th component to fail, 1 ≤ s ≤ (n − r), the maximum Likelihood prediction (point MLPP and interval MLPI), Bayesian prediction (point BPP and interval BPI) can be obtained.
The conditional PDF of T s ≡ T s;n, m,k given that the T r ≡ T r;n, m,k components that had already failed are In the case when m = −1, substituting ( 5) and ( 7) in ( 14), the conditional PDF takes the form In the case when m −1, substituting ( 5) and ( 7) in ( 14), the conditional PDF takes the form Now, we shall study two cases: the first case is when the parameter α is known and the second is when the parameters θ and α are unknown.

Prediction When α j Is Known
Suppose that the mixing proportion, p j , j = 1, 2 and α j , j = 1, 2 are known.

1-Interval prediction:
The MLPI for any future observation t s , s = r + 1, r + 2, ..., n can be obtained by A (1−τ)×100 % MLPI (L, U) of the future observation t s is given by solving the following two nonlinear equations 2-Point prediction: The MLPP for any future observation t s , s = r + 1, r + 2, ..., n can be obtained by replacing the shape parameters θ 1 and θ 2 by θ 1 (ML) and θ 2 (ML) which, obtained from (12).

Bayesian Prediction
Suppose that the mixing proportion, p j and α j are known.Let θ j , j = 1, 2, have a gamma prior distribution with PDF π(θ These are chosen since they are the conjugate priors for the individual parameters.The joint prior density function of θ = (θ 1 , θ 2 ) is given by π where j = 1, 2 θ j > 0, (ν j , β j ) > 0.
It follows, from ( 10) and ( 21), that the joint posterior density function is given by where The Bayes predictive density function can be obtained using ( 15), ( 16) and ( 22) as follows: 1-Interval prediction: Bayesian prediction interval, for the future observation T s;n, m,k , s = r + 1, r + 2, ..., n, can be computed by approximated h * 1 (t s |t) using the MCMC algorithm, see Ahmad et al. (2011) and Ateya (2011), using the form where μ is the number of generated parameters and θ i j , i = 1, 2, 3, ..., μ, they are generated from the posterior density function (22) using Gibbs sampler and Metropolis-Hastings techniques, for more details see Press (2003).
A (1 − τ) × 100 % BPI (L, U) of the future observation t s is given by solving the following two nonlinear equations Numerical methods are generally necessary to solve the above two equations to obtain L and U for a given τ.

2-Point prediction:
1-BPP for the future observation t s based on BSEL function can be obtained using where t s (ML) is the ML prediction for the future observation t s which can be obtained using ( 19) and E(t s |t) can be obtained using 2-BPP for the future observation t s based on BLINX loss function can be obtained using where t s (ML) is the ML prediction for the future observation t s which can be obtained using ( 19) and E(e −at s |t)] can be obtained using

Prediction When α j and θ j Are Unknown
Let the mixing proportion, p j is known and θ j , α j , j = 1, 2, are unknown.

1-Interval prediction:
The MLPI for any future observation t s , s = r + 1, r + 2, ..., n can be obtained by A (1 − τ) × 100 % MLPI (L, U) of the future observation t s is given by solving the two nonlinear equations ( 18).

Bayesian Prediction
Let the mixing proportion, p j is known and θ j , α j , j = 1, 2, are unknown.Suppose that the prior knowledge about the parameters α and θ is adequately represented by the function g(α, θ) given by where g(θ j |α j ) is Gamma (d j , 1 α j ), g(α j ) is Gamma (b j , 1 c j ) with respective densities It then follows, by substituting ( 35) and ( 36) in (34), that the prior PDF of α j and θ j is given by where b j , c j and d j are the prior parameters (also known as hyperparameters).It follows, from ( 10) and ( 37), that the joint posterior density function is given by International Journal of Statistics and Probability Vol. 1, No. 2; 2012 where The Bayes prediction density function of T s ≡ T (s, n, m, k) can be obtained, see Aitchison and Dunsmore (1975), by

1-Interval prediction:
Bayesian prediction interval, for the future observation T s;n, m,k , s = r + 1, r + 2, ..., n, can be computed by h * 2 (t s |t) which can be approximated using the MCMC algorithm, see Ahmad et al. (2011), by the form where θ i j , α i j , i = 1, 2, 3, ..., μ are generated from the posterior density function (38) using Gibbs sampler and Metropolis-Hastings techniques.A (1 − τ) × 100 % BPI (L, U) of the future observation t s is given by solving the following two nonlinear equations Numerical methods are generally necessary to solve the above two equations to obtain L and U for a given τ.

2-Point prediction:
1-BPP for the future observation t s based on BSEL function can be obtained using where t s (ML) is the ML prediction for the future observation t s which can be obtained using ( 33) and E(t s |t) can be obtained using 2-BPP for the future observation t s based on BLINX loss function can be obtained using where t s (ML) is the ML prediction for the future observation t s which can be obtained using ( 33) and E(e −at s |t)] can be obtained using

Prediction in Case of Two-sample Scheme
Based on the informative T 1;n, m,k , T 2;n, m,k , ..., T r;n, m,k GOS drawn from the MTEW distribution and let . ., N, M > 0, K > 0 be a second independent generalized ordered random sample (of size N) of future observations from the same distribution.We want to predict any future (unobserved) where ω b j = (−1) j b−1 j and γ j = K + (N − j)(M + 1).Substituting from ( 5) and ( 7) in ( 48), we have: 4.1 Prediction When α Is Known

1-Interval prediction:
The MLPI for any future observation y b , 1 ≤ b ≤ N can be obtained by A (1−τ)×100 % MLPI (L, U) of the future observation y b is given by solving the following two nonlinear equations 2-Point prediction: The MLPP for any future observation y b can be obtained by replacing the shape parameters θ 1 and θ 2 by θ 1 (ML) and θ 2 (ML)

Bayesian Prediction
The predictive density function of Y b , 1 ≤ b ≤ N is given by: where for M −1 and m −1 Also, when M = −1 and m = −1 1-Interval prediction: Bayesian prediction interval, for the future observation Y b , 1 ≤ b ≤ N, can be computed using ( 55) and ( 56) which can be approximated using MCMC algorithm by the form where θ i j , i = 1, 2, ..., μ are generated from the posterior density function ( 22) using Gibbs sampler and Metropolis-Hastings techniques.
A (1 − τ) × 100 % BPI (L, U) of the future observation y b is given by solving the following two nonlinear equations Numerical methods such as Newton-Raphson are generally necessary to solve the above two nonlinear equations ( 58) and ( 59), to obtain L and U for a given τ.

2-Point prediction:
1-BPP for the future observation y b based on BSEL function can be obtained using where y b (ML) is the ML prediction for the future observation y b which can be obtained using ( 53) and E(y b |t) can be obtained using 2-BPP for the future observation y b based on BLINX loss function can be obtained using where y b (ML) is the ML prediction for the future observation y b which can be obtained using ( 53) and E(e −ay b |t)] can be obtained using

Bayesian Prediction
The predictive density function of Y b , 1 ≤ b ≤ N is given by: For M −1 and m −1 Also, when M = −1 and m = −1 1-Interval prediction: Bayesian prediction interval, for the future observation Y b , 1 ≤ b ≤ N, can be computed using ( 68) and ( 69) which can be approximated using MCMC algorithm by the form where α i j , θ i j , i = 1, 2, ..., μ are generated from the posterior density function (38) using Gibbs sampler and Metropolis-Hastings techniques.
A (1 − τ) × 100 % BPI (L, U) of the future observation y b is obtained by solving the following two nonlinear equations Numerical methods such as Newton-Raphson are necessary to solve the above two nonlinear equations ( 71) and ( 72), to obtain L and U for a given τ.
6) The BPP for the future observation y 1 , is computed based on BSEL function using (60) when α j is known and (73) when α j and θ j are unknown.
7) The BPP for the future observation y 1 , is computed based on BLINX loss function using (62) when α j is known and (75) when α j and θ j are unknown.8) Generate 10, 000 samples each of size N = 5 from a MTEW distribution, then calculate the coverage percentage (CP) of Y 1 .

Conclusions
In our study, we obtained the maximum likelihood prediction (point and interval) and Bayesian prediction (point and interval) for the future observation from mixture of two Exponentiated Weibull (MTEW) distributions.From Tables (1-4) we observe the following: 1) Point and 95% interval predictors for future observations are obtained using a one-sample and two-sample schemes based on a MTEW distribution.Our results are specialized to upper record values.
2) It is evident from all tables that, the lengths of the MLPI and BPI decrease as the sample size increase.
3) The percentage coverage improves by use of a large number of observed values.
4) It may be noticed that when ω = 1, we obtain the MLEs while the case ω = 0, yields the Bayes prediction under SE and LINX loss function.