EXPONENTIATED PARETO DISTRIBUTION : A BAYES STUDY UTILIZING MCMC TECHNIQUE UNDER UNIFIED HYBRID CENSORING SCHEME

Received 4/3/2018 Revised 25/3/2018 Accepted 6/7/2018 Abstract: This article aims to study the problem of point and interval estimations of the exponentiated Pareto distribution utilizing unified hybrid censored scheme (HCS). We utilize three methods, including the maximum likelihood, parametric bootstrap and Bayes of estimating the unknown parameters, reliability, hazard rate functions and coefficient of variation. Furthermore, Markov Chain Monte Carlo samples utilizing importance sampling scheme are utilized to generate the Bayes estimates and the credible intervals for unknown quantities. The findings of Bayes method computed using balanced loss function. The suggested methods can be understood by analysing a set of real data.


Introduction
In life-testing and reliability studies, it is prefered to use a censoring scheme that can balance the total time spent on the experiment, the number of units utilized in the experiment and the efficiency of statistical analysis under the findings of the experiment.Type-I (time) and Type-II (failure) censoring schemes are deemed to be two of the most known censoring schemes.In Type-I, the experimental time is fixed and the number of observed failures is random.On the other hand, in Type-II the number of observed failures is fixed and the experimental time is random.More above, the two types do not allow units removal at certain points, other than the expermient terminal point.In order to overcome the challenge of inflexibility in the traditional Type I and Type II censoring schemes, more comprehensive censoring schemes are proposed.Many authors presented several hybrid censoring schemes (HCS), including Type-I HCS (Epstein [1]), Type-II HCS (Childs et al. [2]) and generalized HCS, where it contains generalized Type-I and Type-II HCS (see, Chandrasekar et al. [3]).Generalized HCS have some drawbacks, for instance in generalized Type-I HCS the experiment terminated when the same time or before T , so we cannot ensure observing r failures.On the other hand, in the generalized Type-II HCS, we cannot observe any failure at all or observe only a few number of failures until the pre-fixed time T 2 .To overcome the problems of these schemes, Balakrishnan et al. [4] presented a unified HCS which can be spicified as follows: Fix integers r, k ∈ {1, ...., n} where k < r < n and T 1 , T 2 ∈ (0, ∞) where T 2 > T 1 .When k th failure occurs before time T 1 , the experiment is terminated at min max{X r:n , T 1 }, T 2 .When k th failure occurs between T 1 and T 2 , the experiment is terminated at min{X r:n , T 2 } and when k th failure occurs after time T 2 , the experiment is terminated at X k:n .Therefore, depending on this censoring scheme, we can ensure that the experiment would be finished at most in time T 2 with at least k failures and if not, we can ensure exactly k failures.Thence, based on this unified HCS, we get the following six cases: Case I: 0 < x k:n < x r:n < T 1 < T 2 , the experiment terminated when T 1 .Case II: 0 < x k:n < T 1 < x r:n < T 2 , the experiment terminated when x r:n .Case III:0 < x k:n < T 1 < T 2 < x r:n , the experiment terminated when T 2 .
Gupta et al. [10] proposed that the exponentiated Pareto distribution (EPD) can be utilized quite effectively in analyzing many lifetime data.Ali et al. [11,12] studied several exponentiated distributions, including, EPD and discussed their properties.They indicated that the EPD is a good fit to the tail-distribution for NASDAQ data.Shawky and Abu-Zinadah [13] examined different estimators of the unknown parameters based on EPD.Afify [14] discussed estimation of the EPD under type I and type II censoring schemes.Finally, Mahmoud et al. [15] explored the estimation of parameters for the EPD depending on progressively type-II right censored data.The probability density function (PDF) and the cumulative distribution function (CDF) of the EPD with two shape parameters β and θ are given, respectively, by and Futher, the reliability R(t) and hazard rate H(t) functions of the EPD(β, θ) at time t are given, respectively, by and Many scientific areas such as engineering, economics, biology, and psychology have utilized the coefficient of variation (CV (X)) in queueing and reliability theory (see, Sharma and Krishna [16]).CV (X) of the EPD(β, θ) is where E(X) and Var(X) are given, respectively, by where B(a, b) = Γ(a) Γ(b) Γ(a+b) is Beta function.According to Equation (1.5), the CV (X) for the EPD(β, θ) is given by

Estimation of the Parameters
In the present section, we estimate β and θ through considering the maximum likelihood estimators (MLEs) as well as R(t), H(t) and CV (X).We calculate the observed Fisher information under the likelihood equations where φ 1 = β , φ 2 = θ for i, j = 1, 2. Also, we compute asymptotic confidence intervals (CIs) for β, θ, R(t), H(t) and CV (X) utilizing the normality property of corresponding MLEs.

Maximum likelihood estimation
Assumeing that X 1:n < X 2:n < . . .< X n:n be a random sample of size n from EPD(β, θ), according to the unified HCS described in Section 1, the likelihood function takes the form where for Case III, (r, x r:n ) , D 1 = 0, . . ., k − 1; for Case VI. (2.2) Where, D m indicates the number of failures till time T m , m = 1, 2. Utilizing Equations (1.2) and (1.1) in Equation (2.1) and ignoring the normalized constant, the likelihood function of β and θ is obtained as 3) The log-likelihood function (x; β, θ) of β and θ is obtained from Equation (2.3) as Differentiating Equation (2.4) with respect to β and θ and equating each result to zero, we obtaine likelihood equations as and Equations (2.5) and (2.6) cannot be solved analytically for β and θ, hence, we utilize numerical methods such as Newton's Raphson method to compute maximum likelihood estimates for β and θ, say βML and θML , respectively.Consequently, utilizing the invariance property, the MLEs of R(t), H(t) and CV (X) can be computed after substituting β and θ by βML and θML , as (2.7)

Approximate Interval Estimation
The Fisher information matrix I(β, θ) is then obtained by expecting where φ 1 = β , φ 2 = θ for i, j = 1, 2. Evidently, obtaining the exact expressions of the above expectation represents a challenge.Accordingly, we extract the approximate asymptotic variance-covariance matrix of MLEs given as Thus, it can be utilized to calculate the 100(1−τ )% approximate CIs for the parameters β and θ which are given, respectively, by βML where Z τ 2 is the percentile of N (0, 1) with right-tail probability τ 2 .Moreover, to construct the asymptotic CIs of the reliability, hazard functions and coefficient of variation, we need to compute the variances of them by utilizing the delta method, see Greene [17].Assume that are the first derivatives of the R(t), H(t) and CV (X) with respect to the parameters β and θ, respectively.The approximate asymptotic variances of RML (t), ĤML (t) and CV ML (X) can be computed, respectively, by where, A T k is the transpose of A k , k = 1, 2, 3.These findings yield the approximate CIs for R(t), H(t) and CV (X) as follows: Var CV ML (X) . (2.11)

Bootstrap confidence intervals
In the present section, two parametric bootstrap procedures introduced to construct the bootstrap CIs of β, θ, R(t), H(t) and CV (X): The percentile bootstrap (Boot-p) confidence interval by Efron [18] and the bootstrap-t (Boot-t) confidence interval proposed by Hall [19].The following algorithms show the estimation of CIs utilizing both methods.

Compute the MLEs depending on the bootstrap sample and indicate this bootstrap estimate by φ
or CV (X) .
6. Duplicate the procedues 2-5, NBoot times, then calculate NBoot in ascending orders, then calculate the following ordered sequences 8. Suppose that G 2 (u) = P (T * ≤ u) be the CDF of T * .For a given u, Subsequently, the approximate bootstrap-t 100(1 − τ )% CIs of φ is given by

Bayes estimation utilizing MCMC technique
In the present section, we compute Bayesian estimates and the corresponding credible intervals of β and θ in addition to some lifetime parameters, including R(t), H(t) and CV (X).We suppose that the parameters β and θ are independent and follow the gamma prior distributions Here, all the hyper parameters a 1 , a 2 , b 1 and b 2 are known and positive.As a consequence, the joint prior distribution π(β, θ) The posterior distribution of the parameters β and θ indicated by π * (β, θ|x) up to proportionality is determined by combining the likelihood function Equation (2.3) with the prior Equation (4.1) via Bayes' theorem, then π * (β, θ|x) is formulated as Consequently, we can determine the Bayes estimate of any parameters function, such as h(β, θ), based on squared error loss function, as It may be noted that, the computation of (4.4) cannot be solved analytically.It is due to the complicated form of the likelihood function given in Equation (2.3).Consequently, MCMC technique is utilized to generate samples from Equation (4.3).Then, we use samples to calculate the Bayes estimate of β and θ and any function of them such as R(t), H(t) and CV (X) and to construct associated CIs.To apply the MCMC technique, we use the Gibbs within Metropolis sampler which requires the derivation of the complete set of conditional posterior distribution.From (4.3), π * (β, θ|x) up to proportionality is formulated as From (4.5), the conditional posterior densities of β and θ are given, respectively, by ) The conditional posterior density of β in Equation (4.6) is gamma density with shape parameter (R+a 1 ) and scale parameter Subsequently, samples of β can be easily generated utilizing any gamma generating routine.The conditional posterior distribution of θ in Equation (4.7) can not be present in standard form, but its plot is similar to normal distribution.So, Gibbs sampling is not a straight-forward option.Using Metropolis-Hasting (M-H) sampler is necessary for applying MCMC methodology.The following procedures show the process of the M-H algorithm within Gibbs sampling: 1.For given t, start with initial guess (β 0 , θ 0 ).
(ii) Compute the acceptance probability .

Bayes estimation utilizing balanced loss functions
Asymmetric loss function is often choosen, since it makes the statistical inferences more practical and applicable.Linearexponential loss function (see Varian [20]) and General entropy loss function (see Calabria et al. [21]) are the most well-known asymmetric loss functions.However, a more generalized loss function named the balanced loss function introduced by Jozani et al. [22] is given by where ρ(Θ, δ) is an arbitrary loss function and δ 0 is chosen as a prior 'target' estimator of Θ, computed for the utilization of the standard of maximum likelihood, least-squares or unbiasedness.Depending on the observed value of δ 0 (X), loss L ρ, ω, δ0 reflects a desire of closeness of δ to the target estimator δ 0 and the unknown parameter Θ.With the relative importance of these criteria governed by the choice of 0 ≤ ω < 1. L ρ, ω, δ0 is specified to different choices of loss function for instance, for absolute value, balanced squared error loss (BSEL), balanced linear-exponential loss (BLINEXL) and balanced general entropy loss (BGEL) functions.
If we put ρ(Θ, δ) = (Θ − δ) 2 in Equation (5.1), it leads to BSEL function, see Ahmadi et al. [23], is then the Bayes estimator of the unknown parameter Θ based on BSEL is Also, if we put ρ(Θ, δ) = e c (δ−h(Θ)) − c (Θ − δ) − 1 with shape parameter c, c = 0 in Equation (5.1), it leads to BLINEXL function, see Zellner [24].Thus, the Bayes estimation of Θ based on BLINEXL function is (5.4) In addition, the BGEL function with shape parameter q is obtained with the choice of ρ(Θ, δ) = ( δ Θ ) q − q ln( δ Θ ) − 1. Thence, the Bayes estimation of Θ based on BGEL function is (5.5) The balanced loss functions are more comprehensive, since they include the MLE and both symmetric and asymmetric Bayes estimates as special cases.For instance, when ω = 1 in Equation ( 5.3) the Bayes estimate based on BSEL function limited to ML estimate and when ω = 0 it limited to the Bayes estimate relative to SEL function (symmetric).As well, the Bayes estimator based on BLINEXL function in (5.4) limited to ML estimate when ω = 1 and if ω = 0 it limited to the case of LINEXL function (asymmetric).Furthermore, the Bayes estimator based on BGEL function reduces to MLE if ω = 1 in Equation (5.5) and when ω = 0, it reduces to the case of general entropy loss function.When Θ = β, θ, R(t), H(t), CV (X) and according to MCMC technique described above, the approximate posterior means under BSEL, BLINEXL and BGEL functions can be formulated, respectively, by ) ) (Θ (j) ) −q −q . (5.8) Subsequently, the approximate Bayes estimates for Θ = β, θ, R(t), H(t) or CV (X) based on BSEL, BLINEXL and BGEL functions can be computed, respectively, by (Θ (j) ) −q −q . (5.11) 6 Application a set of real data We introduce an example utilizing a set of real data to explain the computations of the methods proposed in this article.The set of real data is extracted from Ghazal and Hasaballah [25].The K-S test value is 0.0658573 of the EPD.From Figure 1, it is clearly seen that the EPD is a good model fitting this data.To facilitate the computation processes, we have divided each data point by 10.According to unified HCS, we consider the six cases as following: mode, standard deviation (SD), standard errors (SE) and skewness for β, θ, R(t), H(t) and CV (X) are reported in Table 1.

Concluding remarks
The aim of this article is to utilize various methods (MLEs, Boot-p, Boot-t and MCMC) to estimate and construct CIs of the parameters β and θ also reliability, hazard functions and coefficient of variation of the exponentiated Pareto distribution based on unified hybrid censoring data.We assume that the gamma priors of β and θ and the Bayes estimators based on the assumptions for balanced squared error loss, balanced linear-exponential loss and general entropy loss functions.In this article, the Bayes estimates cannot be computed in closed form.Thus, we utilize the MCMC method to calculate the approximate Bayes estimates and the corresponding CIs.Further, the details have been explained utilizing a real life data example.

Table 1 :
MCMC findings for some posterior characteristics of six cases.

Table 4 :
Bayes MCMC estimates based on BSEL, BLINEXL and BGEL functions for Case I.

Table 5 :
Bayes MCMC estimates based on BSEL, BLINEXL and BGEL functions for Case II.

Table 6 :
Bayes MCMC estimates based on BSEL, BLINEXL and BGEL functions for Case III.

Table 7 :
Bayes MCMC estimates based on BSEL, BLINEXL and BGEL functions for Case IV.

Table 8 :
Bayes MCMC estimates based on BSEL, BLINEXL and BGEL functions for Case V.

Table 9 :
Bayes MCMC estimates based on BSEL, BLINEXL and BGEL functions for Case VI.