Prediction Based on Generalized Order Statistics from a Mixture of Rayleigh Distributions Using MCMC Algorithm

This article considers the problem in obtaining the maximum likelihood prediction (point and interval) and Bayesian prediction (point and interval) for a future observation from mixture of two Rayleigh (MTR) 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. Numerical example is presented in the methods proposed in this paper.

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 [19][20][21][22][23][24][25].
Let the random variable follows Rayleigh lifetime model, its probability density function (PDF), cumulative distribution function (CDF) and reliability function (RF) are given below: (3) Also, the hazard rate function (HRF)    (5) where, for   0 p  1. k p the mixing proportions j and The case of , in (5), is practical importance and so, we shall restrict our study to this case.In such case, the population consist of two sub-populations, mixed with proportions 1 and 2 1   .In this paper, the components are assumed to be Rayleigh distribution whose PDF, CDF, RF and HRF are given, respectively, by            , 1, 2 j (9) where, for   , the mixing proportions j p are such Several authors have predicted future order statistics and records from homogeneous and heterogeneous populations that can be represented by single component distribution and finite mixtures of distributions, respectively.For more details, see [9,10,26].
For Bayesian approach, the performance depends on the form of the prior distribution and the loss function assumed.A wide variety of loss functions have been developed in the literature to describe various types of loss structures.The balanced loss function was suggested by [34].[35] introduced an extended class of the balanced loss function of the form where is a suitable positive weight function and Suppose that 1; , , 2; , , ; , , , , , are the first (out of ) GOS drawn from the mixture of two Rayleigh MTR distribution.The likelihood function (LF) is given in [1], for by where , is the parameter space, and where and are given, respectively, by ( 5) and (7).

 h t
The purpose of this paper is to obtained the maximum likelihood prediction (point and interval) 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, and j  of the MTR distribution based on GOS.The conjugate prior is assumed to carry out the Bayesian analysis.
The results are specialized to the upper record values.The rest of the article is organized as follows.Section 2 deals with the derivation of the maximum likelihood estimators of the involved parameters.Sections 3 and 4, deals with studying the maximum likelihood (point and interval) and the Bayes prediction (point and interval) in the case of one-sample scheme and two-sample scheme.In Section 5, the numerical computations results are presented and the concluding remarks.

Maximum Likelihood Estimation (MLE)
Substituting ( 6), ( 7) in (11), the LF takes the form Take the logarithm of ( 13), we have where  (15) do not yield explicit solutions for and j  , and have to be solved numerically to obtain the ML estimates of the three parameters.Newton-Raphson iteration is employed to solve (15).
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 [19,36,37].

Prediction in Case of One-Sample Scheme
Based on the informative T T  given that the components that had already failed is In the case when , substituting ( 6) and ( 7) in (17), the conditional PDF takes the form And in the case when , substituting ( 6) and ( 7) in (17), the conditional PDF takes the form . , In the following, we considered two cases: the first is when the mixing proportion p is known and the second is when the two parameters  and p are assumed to be unknown.

Prediction When p Is Known
In this section we estimate 1 and 2   , assuming that the mixing proportion, 1 p and 2 p are known.

n p r
1.1.Maximum Likelihood Prediction Maximum likelihood predictio can be obtain using ( 18) and ( 19) by replacing the shape aramete s 1 which is obtained from (15).

1) Interval prediction
The MLPI for any future observation s t , s = r + 1, r + 2, , n  can be obtained by

2) Point prediction
The MLPP for any future observation ,

Bayesian Prediction
When the mixing proportion, p is known.Let the para- These are chosen since they are the conjugate priors for the individual parameters.The joint prior density function of It then follows, from ( 13) and (24), that th ven by  e joint posterior density function is gi where The Bayes predictive density functi on can be obtained using ( 18), ( 19) and ( 25) as follow: Bayesian pred ; , , , where  is the number of generated parameters and i j  , i 1, 2, 3, ,    .They are generated from the posterio ) using Gibbs sampler and Me- Numerical methods are generally necessary to solve the above two equ gi ations to obtain L and U for a ven  .
2) Point prediction a) BPP for the future observation s t based on BSEL

 
function can be obtained using   , are assumed to be unknown.

Maximum Likelihood Prediction
Maximum likelihood prediction can be obtain using (18) ) by and ( 19replacing the parameters p , 1  and 2  by  which we obtained using (15).

1) Interval prediction
The MLPI for any future observation , ) of the future observation s t is given by solving the following two nonline r Equations ( 1).
2) P iction a 2 oint pred e observation , The MLPP for any futur can be obtained by replacing p , 1 the shape parameters  and 2  by A joint prio and for 1, 2 0, Using the likelihood function (13) and the prior density function (38), the posterior density function will be in the form The Bayes prediction density function of see [39], by can be computed by ap-  proximated   s Q t t using the MCMC algorithm, see [24], using the form   6) and ( 7) in (49), we have:

t pR t p h t p h t p R t p R t ph y p h y
where , 1,2,3, ,

G y y G y y
Numerical methods such as Newton-Raphson are genecessary to solve erally n the above two nonlinear Equations ( 59) and (60), to obtain L and U for a given  .
2) P n a) BPP for the future observation b y based on BSEL function ca oint predictio n be obtained using where ˆb ML y is the ML prediction for the future observation b y which can be obtained using (54) and y is given by solving the following two nonlin equations is given by:

ayesian Prediction
The predic function of

Simulation Procedure
In this subsection we will consider the upper record valich c ues wh an be obtained from the GOS by taking

Conclusions
It may be observed: racteristic Property of instance from the criterion of maximum likelihood (ML), least squares or unbiasedness among others.They give a general Bayesian connection between the case of and ,U) of the future observais given by solving the following two nonlinear (52) A tion y b equations for the future observation b y based on BLINX loss function can be obtained using from the posterior density function(39) using Gibbs sampler and Metropolis-Hastings techniques.
  methods such as Newton-Raphson are necessary to solve the above two nonlinear equations (72) and (73), to obtain L and U for a given  .2) Point prediction a) BPP for the future observation b y ba function can be obtained using of b Y .The computational (our) results ere co puted by using Mathematica 7.0.When p is know the prior parameters chosen point and 95% interval predictors for the future upper record value are computed in case of the on 2311-2318.doi:10 [3] M. Ahsanullah, "Generalized Order Statistics from Exponential Distributiuon," Journal of Statistical Planning and Inference, Vol.85, No. 1-2, 2000, pp.85-91.doi:10.1016/S0378-3758(99)00068-3[4] U. Kamps and U. Gather, "Cha e-and two sample predictions, respectively.