Estimation Based on Generalized Order Statistics from a Mixture of Two Rayleigh Distributions

This article is concerned with the problem of estimating the parameters, reliability and hazard rate functions of the mixture of two Rayleigh distributions (MTRD) based on generalized order statistics (GOS ). The maximum likelihood and Bayes methods of estimation are used for this purpose. The Markov chain Monte Carlo (MCMC) method is used for obtaining the Bayes estimates under the squared error loss and LINEX loss functions. Our results are specialized to progressive Type-II censored order statistics and upper record values. Comparisons are made between Bayesian and maximum likelihood estimators via a Monte Carlo simulation study.


Introduction
The Rayleigh distribution (RD) was first derived by Lord Rayleigh in connection with a study of acoustical problems.Since then many investigators have used the RD or some related forms of it in a variety of engineering, wave propagation, radiation and analysis of target data studies.The RD is also used to model wave heights in oceanography, and in communication theory to describe hourly median and instantaneous peak power of received radio signals.Several such situations have been discussed by Polovko (1968), Takeshi Yamane (1998), Zhi Ren et al. (2011), and many others.The RD is a special case of two parameter Weibull distribution.A random variable T is said to have a RD with parameter θ if its probability density function (PDF) is given by f (t) = 2θ t e −θt 2 , t > 0, (θ > 0). (1) The cumulative distribution function (CDF), reliability function (RF) and the hazard rate function (HRF) are given, respectively, by where . Mixtures of distributions arise frequently in life testing, reliability, biological and physical sciences.Some of the most important references that discussed different types of mixtures of distributions are a monograph by Everitt and Hand (1981), Titterington et al. (1985) and McLachlan and Basford (1988).Bayesian inferences based on finite mixture distribution have been discussed by several authors.Papadapoulos and Padgett (1986) considered Bayesian estimation of the mixing parameter, mean and reliability function of a mixture of two exponential lifetime distributions based on right censored samples.Attia (1993) considered the MT RD and obtained estimates of model parameters using maximum likelihood (ML) and Bayesian approach with censored sampling.Ahmad et al. (1997) derive approximate Bayes estimation for mixture of two Weibull distributions under Type-II censoring.Jaheen (2005b) considered estimation for the mixed exponential distribution based on record statistics.Soliman (2006) obtained the estimates of the parameters and functions of these parameters of the MT RD based on progressively Type-II censored samples when the mixing proportion p is known.Saleem and Aslam (2008a & b) use ordinary type I right censored data for Bayesian analysis of Rayleigh mixture.Saleem and Irfan (2010) studied some properties of the Bayes estimates of the Rayleigh mixture parameters.The PDF, CDF, RF and HRF of the MT RD are given, respectively, by where, for j = 1, 2, the mixing proportions p j are such that 0 ≤ p j ≤ 1, p 1 + p 2 = 1 and f j (t), F j (t), R j (t) are given from ( 1), ( 2), (3) after using θ j instead of θ.
The property of identifiability is an important consideration on estimating the parameters in a mixture of distributions.Also, testing hypothesis, classification of random variables, can be meaning fully discussed only if the class of all finite mixtures is identifiable.Identifiability of mixtures has been discussed by several authors, including Teicher (1963), AL-Hussaini and Ahmad (1981) and Ahmad (1988).
The purpose of this paper is to estimate the parameters, RF and HRF of the MT RD based on GOS using ML and Bayes methods.The non-informative prior and the conjugate prior are assumed to carry out the Bayesian analysis.The results are specialized to progressive Type-II censored order statistics and record values.This paper is organized as follows: In Section 2, the ML estimators of parameters, RF and HRF of the MT RD are derived.In Section 3, the MCMC method is used for obtaining the Bayes estimators of parameters, RF and HRF of the MT RD under the squared error loss and LINEX loss functions.Comparisons between Bayesian and ML estimators via Monte Carlo simulation study are made in Section 4. Finally, Concluding remarks about comparisons between the estimators are considered in Section 5.
Differentiating (7) with respect to the parameters p and θ j and equating to zero gives the following likelihood equations where, for j = 1, 2 Equations ( 8) do not yield explicit solutions for p and θ j , j = 1, 2, and have to be solved numerically to obtain the ML estimates of the three parameters, Newton-Raphson iteration is employed to solve (8).The corresponding ML estimates of the reliability function R(t) and the Hazard rate function H(t) are given respectively by ( 3) and (4) after replacing p, θ 1 and θ 2 by their ML estimates p, θ1 and θ2 , (the solution of the above nonlinear equations).

Prior, Posterior and Bayes Estimators
Recently, there has been a considerable amount of interest in the Bayesian approach in estimation and reliability studies.It has received frequent attention for analyzing failure data and other time-to-event data, and has been often proposed as a valid alternative to traditional statistical perspectives.The Bayesian approach to estimation of the parameters and reliability analysis allows prior subjective knowledge on lifetime parameters and technical information on the failure mechanism, as well as experimental data, to be incorporated into the inferential procedure.Bayesian methods usually require less sample data to achieve the same quality of inferences than methods based on sampling theory.In this section, we present the Bayesian estimation for the parameters, RF and HRF for MT RD based on GOS .In this section, Bayesian estimation for the parameters of the MT RD is considered under squared error and LINEX (Linear-Exponential) loss functions.

Bayes Estimation Using Conjugate Prior
Let p, θ 1 and θ 2 are independent random variables such that p ∼ Beta(b 1 , b 2 ) and for j = 1, 2, θ j to follow an inverted gamma prior distribution with PDF A joint prior density function of θ = (p, θ 1 , θ 2 ) is then given by where 0 < p 1 < 1, p 2 = 1 − p 1 and for j = 1, 2 θ j > 0, (b j , α j , β j ) > 0. It follows, from ( 6) and ( 11), that the joint posterior density function is given by where Under the squared error loss and LINEX loss functions, the Bayes estimator of a function, say φ ≡ φ(p, θ 1 , θ 2 ), are given, respectively, by where the integral is taken over the three dimensional space and a 0. To compute the integral we propose to consider MCMC methods.
The conditional posterior distribution of the parameters p, θ 1 and θ 2 using conjugate prior can be computed and written, respectively, by

Bayes Estimation Using Non-informative Prior
Assuming that all of the parameters consisting θ are positive and independent, and that we are indifferent about the prior information about θ so that we set improper non-informative prior to θ j , j = 1, 2, and p as follows The posterior density function can be obtained from ( 6) and ( 19), as Under the squared error loss and LINEX loss functions, the Bayes estimator of a function, say φ ≡ φ(p, θ 1 , θ 2 ), are given, respectively, by where the integral is taken over the three dimensional space and a 0. To compute the integral we propose to consider MCMC methods.
The conditional posterior distribution of the parameters p, θ 1 and θ 2 using non-informative prior can be computed and written, respectively, by

Numerical Computations
A comparison between ML and Bayes estimators, under either a squared error or a LINEX loss functions, is made using a Monte Carlo simulation study in the two following cases:

Progressive Type-II Censored Order Statistics
The progressive Type-II censored order statistics can be obtained from the GOS as a special case by taking m i = r i for i = 1, 2, ..., m − 1 and k = r m + 1.Therefore, the estimation results obtained in the above sections can be specialized to the progressive Type-II censored order statistics.Estimates of parameters, RF and HRF are computed and compared based on Monte Carlo simulation study according to the following steps: 1) For given values of p, θ 1 and θ 2 , we generate progressively Type-II censored samples from the MT RD by using the algorithm described in Balakrishnan and Sandhu (1995), as follows: • using p, θ 1 and θ 2 , with different chooses of n, m and k.We take in our consideration that the progressive censored order statistics T R 1:m:n:k , T R 2:m:n:k , ..., T R m:m:n:k is a progressively Type-II censored sample from a population with CDF (6).
• determine the values of the censored scheme r i , for i = 1, 2, ..., m.
.., U m is the progressively Type-II right censored sample from the Uniform (0,1) distribution.
• for given values of parameters θ 1 , θ 2 and mixing proportion p, set: For i = 1, 2, ..., m, the values of t i for each U i can be obtained numerically from (32).Then the resulting set t 1 , t 2 , ..., t m is the required progressively Type-II right censored sample from the MT RD.
Using the algorithm described above, a progressively Type-II censored samples of size m with different censored schemes are randomly generated from sample of size n simulated from the MT RD.
2) The ML estimates of the parameters p, θ 1 and θ 2 are obtained by solving the nonlinear equations ( 8), with m i = r i , k = r m + 1, i = 1, 2, ..., m − 1, numerically.3) Based on squared error and LINEX loss functions the Bayes estimates of the parameters, reliability and Hazard rate functions are computed, from ( 26) and ( 27), according to the above MCMC method.The above steps are repeated 500 times.The estimated risks (ER) are computed by averaging the squared deviations over the 500 repetitions.

Upper Record Values
The upper record values can be obtained from the GOS by taking m i = −1 for i = 1, 2, ..., n − 1 and k = 1.In this case, ML and Bayes estimates of parameters, RF and HRF are computed and compared based on Monte Carlo simulation study according to the following steps: 2) The ML estimates of the parameters p, θ 1 and θ 2 are obtained by solving the nonlinear equations ( 8), with m i = −1, k = 1, numerically.3) Based on squared error and LINEX loss functions the Bayes estimates of the parameters, reliability and Hazard rate functions are computed, from ( 26) and ( 27), according to the above MCMC method.The above steps are repeated 1000 times.The estimated risks (ER) are computed by averaging the squared deviations over the 1000 repetitions.The computational (our) results were computed by using Mathematica 6.0.Under conjugate prior the prior parameters chosen as b 1 = 1.2, b 2 = 2.3, α 1 = 2, β 1 = 0.3, α 2 = 2, β 2 = 3 which yield the generated values of p = 0.391789, θ 1 = 0.307317 and θ 2 = 3.33166 (as the true values).The true values of R(t) and H(t) when t = 0.5, are computed to be R(0.5)= 0.627253 and H(0.5) = 1.58232.While, under non-informative prior values of (p, θ 1 , θ 2 ) chosen as (0.4, 0.3, 3).The true values of (R(0.5),H(0.5)) are computed to be (0.654517, 1.46916).The value of the shape parameter a of the LINEX loss function is a = (−2, 0.02, 2).The estimated risks (ER) are displayed in Tables 1-2-3-4   1) All of the results obtained in this article in case of Type II right progressively censoring can be specialized to both the complete sample case by taking (m = n, r i = 0, i = 1, 2, 3, 4, ...m), and the Type-II right censored sample for (r i = 0, i = 1, 2, 3, 4, ...m − 1, r m = n − m).
2) From Table 1, we see that the Bayes estimators relative to asymmetric loss functions (LINEX), are sensitive to the value of the shape parameters a, for small a the Bayes estimates under the LINEX loss function (BL), are very close to the Bayes estimates under the squared error loss function.The Bayes estimates of RF under the LINEX loss function (BL), when a = −2 have the smallest ER s as compared with their corresponding estimates.While, the Bayes estimates of θ 1 , θ 2 and HRF have the smallest ER s as compared with their corresponding ML estimates.
In most of the considered cases, it is observed that the ML estimates of p have the smallest ER s as compared with their corresponding Bayes estimates.
3) Also from Table 1, as the proportion m/n increases, the ER s reduce significantly.We observed that for large sample size, the symmetric and asymmetric Bayes estimates are better than the ML estimates.
4) Table 2 shows that the Bayes estimates of θ 1 , θ 2 and HRF under the LINEX loss function have the smallest ER s as compared with their corresponding estimates.While, the Bayes estimates of RF under the squared error loss function have the smallest ER s as compared with their corresponding estimates.In most of the considered cases, Table 2 shows that the ML estimates of p have the smallest ER s as compared with their corresponding Bayes estimates.

Figure 1 .
Figure 1.Estimated Risks (ER) of the estimates of θ = (p, θ 1 , θ 2 ), R(t) and H(t) based on upper record values . Figures1 and 2represents the estimated risks of the estimates in the case of upper record values.

Table 1 .
(Progressively censored samples) ER of the ML estimates and the Bayes (BS , BL) estimates assuming the conjugate prior of p, θ 1 , θ 2 , R(0.5) and H(0.5) for different m, n and schemes BL ≡ Bayes (LINEX loss function); BS ≡ Bayes (squared error loss function).

Table 3 .
(Upper record values) ER of the ML estimates and the Bayes (BS , BL) estimates assuming the conjugate prior of p, θ 1 , θ 2 , R(0.5) and H(0.5) for different sample size BL ≡ Bayes (LINEX loss function); BS ≡ Bayes (squared error loss function).