Asymptotic Optimal Empirical Bayes Estimation of the Parameter of ЭРланга Distribution

This paper aims to study the empirical Bayes estimation of the parameter of ЭРланга distribution under a weighted squared error loss function. Bayes estimator is firstly to derive based on pivot method. Then empirical Bayes estimator of unknown parameter is constructed in a priori unknown circumstances. The asymptotically optimal property of this empirical Bayes estimator is also discussed. It is shown that the convergence rates of the proposed empirical Bayes estimator can arbitrarily close to O(n)) under suitable conditions.


Introduction
Empirical Bayes method is firstly proposed by Robbins in 1956, and since then it becomes a very important statistical inference, which be applied to many fields, such as reliability, lifetime prediction, medical research, ect. [1][2][3][4][5][6]. Empirical Bayes estimations of various statistical models have received great attention. For example, Fan et al. [7] constructed a empirical Bayes estimator of the scale exponential family in the case of identically distributed and positively associated samples under weighted square loss function.
Rousseau and Szabo [8] considered the asymptotic behavior of the marginal maximum likelihood empirical Bayes posterior distribution in general setting. Wang and Wei [9] studied the empirical Bayes estimation problem for the scale-exponential family with errors in variables under the weighted square loss. Liu and Cao [10] discussed the empirical Bayes estimation about location parameter of twoexponential distribution under a LINEX loss function.in case of the negatively associated samples with kernel density method to estimate marginal probability density function. Zhang and Wei [11] first derived the Bayes estimators of variance components for one-way classification random effect model under the weighted square loss function, and then they constructed the empirical Bayes estimators by the kernel estimation method. Seal and Hossain [12] not only used EM algorithm to compute empirical Bayes estimates for different hyperparameters, but also investigated the robustness of empirical Bayes procedure. Steorts and Ghosh [13] considered benchmarked empirical Bayes estimators under the basic area-level model of Fay and Herriot while requiring the standard benchmarking constraint. Jiang [14] studied a monotone regularized kernel general empirical Bayes method for the estimation of a vector of normal means. Park [15] discussed the problem of simultaneous Poisson mean vector estimation and studied the performance of nonparametric empirical Bayes estimator from the view point of risk consistency.
ЭРланга distribution is an important distribution to model the repair time and guarantee the distribution delay time [16]. Pan et al. [17] studied the interval estimation and hypothesis test problem of ЭРланга distribution in case of small sample. Long [18] discussed the estimation of Зрланга distribution based on missing data.
Assume that random variable X has the ЭРланга distribution with the following probability density function and cumulative distribution function, respectively: Lemma 1 let 1 , , n X X ⋯ be the independent identical distribution random samples distributed with ЭРланга distribution (1). Define Then (i) 1 ( ) n p x and 2 ( ) n p x are the unbiased estimator of 1 ( ) p x and 2 ( ) p x ,respectively. That is By Eq.(10) and Eq. (14), = ⋯ is also an independent identical distribution random sequence. Then is an independent identical distribution random sequence. Then

Asymptotic Optimal Properties of the Emprical Bayes Estimator
This section will study the asymptotic optimality property of the proposed empirical Bayes estimator. Under the weighted squared error loss, the Bayes risks of the proposed empirical Bayes estimator ( ) n x In the following discussion, we always assume 1 2 , , ,... c c c represent different constants, even they occur in the same expression they also may take different values.
Lemma 2 [25] Assume that δ is an arbitrary estimator of parameter θ , then Then by Lemma 1, we have Obviously that

∫ ∫
Then the case (ii) is true. Thus we finish the proof of the theorem.

Conclusions
For the estimation of the parameter of ЭРланга distribution, this paper puts forward an empirical Bayes approach to estimate the parameter under a weighted squared error loss function when the prior distribution of the parameter is unknown.. The asymptotically optimal property of the proposed empirical Bayes estimator is also discussed. The convergence rates of the proposed empirical Bayes estimator can arbitrarily close to 1 ( ) n − Ο under suitable conditions. The proposed method can be similarly extended to the construction of empirical Bayesian estimator of other distributions, such as Rayleigh distribution, Lomax distribution, etc.