Exponentiated Rayleigh Distribution: A Bayes Study Using MCMC Approach Based on Unified Hybrid Censored Data

6863 | P a g e J a n u a r y 2 0 1 7 w w w . c i r w o r l d . c o m Exponentiated Rayleigh Distribution: A Bayes Study Using MCMC Approach Based on Unified Hybrid Censored Data M. G. M. Ghazal, H. M. Hasaballah 1 Mathematics Department, Faculty of Science, Minia University El-Minia, Egypt. eghazal39@yahoo.com 2 Mathematics Department, Faculty of Science, Minia University El-Minia, Egypt. hasaballahmohamed@yahoo.com ABSTRACT


INTRODUCTION
Supposing there are n identical units put to test in the same time, when the experiment is terminated at apre-fixed time T, we refer to Type-I censoring scheme. The disadvantage of Type-I censoring scheme is very few failures may occur before time T. The experiment is terminated when the r th failures occur, we refer to Type-II censoring scheme. Furthermore, the disadvantage of Type-II censoring scheme is that the termination time is unknown prior to the experiment. For more details on Type-I and Type-II censoring, and related inferential issues, see Nelson [23] and Balakrishnan and Cohen [4]. A hybrid censoring scheme (HCS) is a mixture of Type-I and Type-II censoring schemes. This HCS was introduced by Epstein [10], to avoid the disadvantages of Type-I and type-II censored scheme, and it has been used in reliability acceptance test in MILSTD-781-C [21], see also Jeong et al. [16], Gupta and Kundu [12], Park and Balakrishnan [24]. Epstein [10] is considered a hybrid censored scheme in which the life-testing experiment is terminated at a random time T*=min{Xr:n, T}, this refer to Type-I HCS. The disadvantage of Type-I HCS is very few failures occurring until the pre-fixed time T. To overcome this disadvantage, Childs et al. [8] introduced a substitute Type-I HCS that would terminate the experiment at the random time T*=max {Xr:n, T}. This HCS is called Type-II HCS. In the same respect, to avoid the disadvantages in these schemes, Chandrasekar et al. [7] proposed two new schemes which are called generalized Type-I and Type-II HCS. In generalized Type-I HCS, fix k, r  (1, 2, …, n) and T  (0,∞) such that k < r < n. If the k th failure occurs before time T, the experiment is terminated at min {Xr:n; T}. If the k th failure occurs after time T, the experiment is terminated at Xk:n, so, it is clear that this HCS modifies the Type-I HCS by allowing the experiment to continue after time T if very few failures had observed until that time point. In generalized Type-II HCS, fix r  (1, 2, …, n) and T1, T2  (0,∞) such that T2 > T1. If the r th failure occurs before time point T1, the experiment is terminated at T1. If the r th failure occurs between T1 and T2, the experiment is terminated at Xr:n. If the r th failure occurs after T2, the experiment is terminated at T2. There are some drawbacks in generalized HCS, such as, in generalized Type-I HCS. Moreover, because the experiment is terminated at the same time or before T, we cannot guarantee observing r failures. While in the generalized Type-II HCS, we cannot observe any failure at all or observe only few number of failures until the pre-fixed time T2. To avoid these drawbacks, Balakrishnan et al. [5] introduced an unified hybrid censoring scheme (UHCS), which can be described as follows, fix r, k  {1, …, n} where k < r < n and T1, T2  (0,∞) where T2 > T1. If the k th failure occurs before time T1, the experiment is terminated at min {max {Xr:n, T1}, T2}. If the k th failure occurs between T1 and T2, the experiment is terminated at min{Xr:n, T2}. and if the k th failure occurs after time T2, the experiment is terminated at Xk:n. Under this censoring scheme, we can guarantee that the experiment would be completed at most in time T2 with at least k failure and if not, we can guarantee exactly k failures.

The Survival function S(t) is given by
The hazard function H(t) is given by The rest of the paper is organized as follows: In Section 2, we discussed the MLEs of the unknown parameters in addition to S(t) and H(t). In Section 3, credible intervals based the MLEs are presented. In Section 4, confidence intervals were obtained from two parametric bootstrap procedures. In Section 5, we apply MCMC technique to obtain the confidence intervals. One data set has been analyzed for illustrative purposes in Section 6. Finally, conclusions are given in Section 7.

MAXIMUM LIKELIHOOD ESTIMATION
In this section, we obtained the MLEs of ER (α, β) distribution when α and β are unknown. Let (x1, . . . , xn) be a random sample of size n from ER(α, β) distribution, then the likelihood function for six cases of the UHCS is as follows: Where R indicates the number of the total failures in experiment up to time C (the stopping time point) and d1 and d2 indicate the number of failures that occur before time points T1 and T2, respectively. From (1.1), (1.2) and (2.1) we get I S S N 2 3 4 7 -1 9 2 1 V o l u m e 1 2 N u m b e r 1 2 J o u r n a l o f A d v a n c e s i n M a t h e m a t i c s 6865 | P a g e J a n u a r y 2 0 1 7 The logarithm of the likelihood function can be written as Applying the first derivative with respect to α and β in (2.3) and equating by zero, then we get the two normal equations as follows where R and C are defined above. The MLEs of α and β can be found by solving the system of equations (2.4) and (2.5), even if the suggested estimators cannot be expressed in closed forms, we can use a suitable numerical technique to obtain the estimators. Moreover, we can obtain the MLEs of S(t) and H(t) after replacing α and β by their MLEs ˆ and ˆ as following

CONFIDENCE INTERVAL
The asymptotic variance-covariance of the MLEs for parameters α and β are given by elements of the inverse of the Fisher information matrix are defined as Unluckily, we found difficult to solve the above expectations. Therefore, we gave the approximate asymptotic variancecovariance matrix for the MLEs, which, can be obtained by I S S N 2 3 4 7 -1 9 2 1 V o l u m e 1 2 N u m b e r 1 2 J o u r n a l o f A d v a n c e s i n M a t h e m a t i c s 6866 | P a g e J a n u a r y 2 0 1 7 w w w . c i r w o r l d . c o m The asymptotic normality of the MLEs can be used to compute the approximate confidence intervals (ACI) for parameters α and β. Therefore, (1-)100% confidence intervals for parameters α and β become Where Z /2 is a standard normal variate.
Moreover; to find the ACI of the S(t) and H(t), we need to find the variances of them. To find the approximate estimates of the variance of Ŝ(t) and Ĥ(t) we use the delta method. The delta method is a general approach for computing confidence intervals for functions of MLEs. Depending on this method, the variance of Ŝ(t) and Ĥ(t) respectively given by Ĥ(t) are the gradient of Ŝ(t) and Ĥ(t) respectively, with respect to α and β. And Vˆ=I -1 (α,β). Therefore, (1-)100% confidence intervals for S(t) and H(t) become and

BOOTSTRAP CONFIDENCE INTERVALS
In this section, two parametric bootstrap procedures are provided to construct the bootstrap confidence intervals of α, β, S(t) and H(t). The first one is the percentile bootstrap (Boot-p) confidence interval proposed by Efron [9]. The second one is the bootstrap-t (Boot-t) confidence interval proposed by Hall [13]. The algorithms for estimating the confidence intervals using both methods are illustrated as follows.
Repeat steps 3 and 4 N Boot times, and obtain , ,...., , in an ascending order to obtain the bootstrap sample  .
6. Repeat step 3 and 5 NBoot times and obtain . ,...., , , in an ascending orders and obtain the ordered sequences   Then the

BAYES ESTIMATION
In this section, we described how to obtain the Bayes estimators and the corresponding credible intervals of parameters α, β, S(t) and H(t) of ER(α, β) distribution under squared error loss, LINEX loss and general entropy loss functions, based on UHCS when α and β are supposed to be unknown. We assume that α and β have the following gamma prior distributions I S S N 2 3 4 7 -1 9 2 1 V o l u m e 1 2 N u m b e r 1 2 J o u r n a l o f A d v a n c e s i n M a t h e m a t i c s 6868 | P a g e J a n u a r y 2 0 1 7 Here all the hyper parameters a1; a2; b1 and b2 are assumed to be known and non-negative.
The joint prior distribution for α and β is Using the joint prior distribution of α and β, we obtained the joint distribution of the data, α and β as Based on (5.15), the joint posterior density function of α and β can be written as Therefore, the Bayes estimator of a function u(α, β) under the squared error loss function is as follows The Bayes estimator of a function u(α,β) under the LINEX loss function is as follows The Bayes estimator of a function u(α,β) under the General entropy loss function is as follows It is very difficult to compute the equations (5.17), (5.18) and (5.19) analytically. Then, we suggested using MCMC to generate samples from (5.20) and approximate these equations under square error loss, LINEX loss and general entropy loss functions. The MCMC method is used for computing the Bayes estimates of the parameter α and β in addition to S(t) and H(t) and also to construct the corresponding credible intervals based on the generated posterior sample. For more details about the MCMC methods see, for example, Upadhyay et al. [33] and Upadhyay and Gupta [34]. We supposed the Gibbs with in Metropolis sampler, which requires the derivation of the complete. From Also the posterior density function of β given α can be written as Therefore, the posterior density function of α given β is gamma with the shape parameter (α1+R) and scale parameter and, therefore, samples of α can be easily generated using any gamma generating routine.
Furthermore, the conditional posterior distribution of β given α in (5.22) cannot be reduced analytically to well known distributions and therefore it is impossible to sample directly by standard methods. So, to generate random numbers from this distribution, we use the Metropolis-Hastings method with normal proposal distribution, see Metropolis et al. [20]). Now, we suppose the next MCMC algorithm to draw samples from the posterior density (5.20) and in turn to obtain the Bayes estimates α and β and any function of them such as S(t) and H(t) and the corresponding credible intervals.
Algorithm of MCMC method:

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Step 5. Compute S(t) and H(t) as

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Step 7. Repeat steps 3-6 N times and obtain α (j) , β (j) , S (j) (t) and H (j) (t), j = M+1, , N.  This data was produced by the national climatic data center (NCDC) in Asheville in the United States of America. Now, one of the most important subjects is type of distribution of any set of data will be known during statistical tests which are called the goodness of fit. We depended on Kolmogorov-Smirnov test to fit whether the data distribution as ER (α, β) distribution or not. The calculated value of the K-S test is 0.0983022 for the ER (α, β) distribution and this value is smaller than their corresponding values expected at 5% significance level, which is 0.13403 at n = 100. We have just plotted the empirical S(t) and the fitted S(t) in Figure1a. Observe that the ER (α, β) distribution can be a good model fitting this data. In Figure1b it shows that all the points of a Q-Q plot are inside the unit square, so, it can be seen that the ER (α, β) distribution fits the data very well. P-value = 0.27052, therefore, the high p-value indicates that ER (α, β) distribution can be used to analyze this data set. Now, we consider the case when the data are censored. We have six cases as following:  In all the six cases, we estimated the unknown parameters, we computed different estimates of the parameters α and β in addition to S(t) and H(t). MLEs, Bootstrap confidence intervals (Boot-p, Boot-t) based on 1000 Bootstrap samples and Bayes estimates using MCMC method based on 11000 MCMC sample and discard the first 1000 values as `burn-in'. To compute the MLEs, we used the numerical method and also computed the 95% ACI; the results are given in Table (1). Also, we computed the 95% ACI Bootstrap confidence intervals. The results are given in Table (2). To compute the Bayes estimates we considered the square error loss, LINEX loss and general entropy loss functions. Under non-informative priors the hyper parameters are a1 = a2 = b1 = b2 = 0. The results are given in Table (3). Under the informative priors the hyper parameters are a1 = 0.9, a2 = 0.8, b1 = 0.5 and b2 = 1.0. The results are given in Table (4). Also, we computed the 95% ACI based on the MCMC samples. The results are given in Table (5). The characteristics of the unknown parameters for MCMC method under non-informative and informative priors are given in Table (6) and Table ( H(t). The histogram of α, β, S(t) and H(t) are given in Figure (3). In all the cases α = 2 and β = 0.026 are considered.    V o l u m e 1 2 N u m b e r 1 2 J o u r n a l o f A d v a n c e s i n M a t h e m a t i c s 6873 | P a g e J a n u a r y 2 0 1 7 w w w . c i r w o r l d . c o m            V o l u m e 1 2 N u m b e r 1 2 J o u r n a l o f A d v a n c e s i n M a t h e m a t i c s 6878 | P a g e J a n u a r y 2 0 1 7 w w w . c i r w o r l d . c o m

CONCLUSION
In this paper, we considered the Bayes estimation of the unknown parameters of the exponentiated Rayleigh distribution when the data is collected under the unified hybrid censored data. The MLEs, the bootstrap confidence intervals and the credible intervals based on the observed Fisher information matrix have been discussed. We supposed the gamma priors for both the unknown parameters and provided the Bayes estimators under the assumptions of squared error loss, LINEX loss and general entropy loss functions. In our study, the Bayes estimates cannot be obtained in explicit form. So, we used the MCMC technique to compute the approximate Bayes estimates and the corresponding credible intervals. We have applied the developed techniques on a real data set.