Minimax Estimation of the Parameter of ЭРланга Distribution Under Different Loss Functions

The aim of this article is to study the estimation of the parameter of ЭРланга distribution based on complete samples. The Bayes estimators of the parameter of ЭРланга distribution are obtained under three different loss functions, namely, weighted square error loss, squared log error loss and entropy loss functions by using conjugate prior inverse Gamma distribution. Then the minimax estimators of the parameter are derived by using Lehmann’s theorem. Finally, performances of these estimators are compared in terms of risks which obtained under squared error loss function.


Introduction
In reliability and supportability data analysis field, the most commonly used distribution are the exponential distribution, normal distribution and Weibull distribution, etc. But in some practical application, such as the repair time, guarantee the distribution delay time, the above several distributions does not just as one wish. At this time ЭРланга distribution was proposed as a suitable alternative distribution [1].
Suppose that the repair time T obeys the ЭРланга distribution with the following probability density function (pdf) and distribution function respectively: Here, θ is the unknown parameter. It is easily to see that 1 ET θ − = , and then the parameter 1 θ − is also often referred to as the mean time to repair equipment. Lv et al. [1] studied the characteristic parameters, such as mean, variance and median and the maximum likelihood estimation of ЭРланга distribution was also derived. Pan et al. [2] studied the interval estimation and hypothesis test of ЭРланга distribution based on small sample, and the difference of exponential distribution with З рланга distribution was also discussed. Long [3] studied the estimation of the parameter of З рланга distribution based on missing data. Yu et al. [4] used the Эрланга distribution to fit the battlefield injury degree, and established the simulating model, then proposed a new method to solve the problem in the production and distribution of battlefield injury in campaign macrocosm. Long [5] studied the Bayes estimation of Эрлангa distribution under type-II censored samples on the basis conjugate prior, Jeffreys prior and no information prior distributions.
The minimax estimation was introduced by Abraham Wald in 1950, and then minimax approach has received great attention and application many aspects by researchers [6][7][8][9]. Minimax estimation is one of the most aspect in statistical inference field. Under quadratic and MLINEX loss functions, The references [10][11][12][13] studied the minimax estimation of the Weibull distribution, Pareto distribution and Rayleigh distributions and Minimax distribution, respectively. Rasheed and Al-Shareefi [14] discussed the minimax estimation of the scale parameter of Laplace distribution under squared-log error loss function. Li [15] studied the minimax estimation of the parameter of exponential distribution based on record values. Li [16] obtained the minimax estimators of the parameter of Maxwell distribution under different loss functions.
The purpose of this paper is to study maximum likelihood estimation (MLE) and Bayes estimation of the parameter of ЭРланга distribution. Further, by using Lehmann's theorem we derive minimax estimators under three loss functions, In Bayesian statistical analysis, loss function plays an important role in the Bayes estimation and Bayes test problems. Many loss function are proposed in Bayesian analysis, and squared error loss function is the most common loss function, which is a symmetric loss function. In many practical problems, especially in the estimation of reliability and failure rates, symmetric loss may be not suitable, because it is to be thought the overestimation will bring more loss than underestimation [17]. Then some asymmetric loss functions are developed. For example, Zellner [18] proposed the LINEX loss in Bayes estimation, Brown [19] put forward the squared log error loss function for estimating unknown parameter, Dey et al. [20] proposed the entropy loss function in the Bayesian analysis.
In this paper, we will discuss the Bayes estimation of the unknown parameter of ЭРланга distribution under the following loss functions: (i) Weighted squared error loss function Under the weighted squared error loss function (7), the Bayes estimator of θ is (ii) Squared log error loss function Squared log error loss function is a asymmetric loss function, which first proposed by Brown for estimating scale parameter. This loss function can also be found in Kiapoura and Nematollahib [21] with the following form: (iii) Entropy loss function In many practical situations, it appears to be more realistic to express the loss in terms of the ratio θ θ . In this case, Dey et al. [20] pointed out a useful asymmetric loss function named entropy loss function: Whose minimum occurs at δ θ = . Also, this loss function has been used in Singh et al. [22], Nematollahi and Motamed-Shariati [23]. The Bayes estimator under the entropy loss (11) is denoted byˆB E θ , obtained by In this section, we will estimate the unknown parameter θ on the basis of the above three mentioned loss functions. We further assume that some prior knowledge about the parameter θ is available to the investigation from past experience with the ЭРланга model. The prior knowledge can often be summarized in terms of the so-called prior densities on parameter space of θ . In the following discussion, we assume the following Jeffrey's noninformative quasi-prior density defined as, Hence, 0 d = leads to a diffuse prior and 1 d = to a noninformative prior. Let be a sample drawn from ЭРланга distribution with pdf (1), and is the observation of X . Combining the likelihood function (3) with the prior density (13), the posterior probability density of θ can be derived using Bayes Theorem as follows Theorem 1. Let be a sample of Э Рланга distribution with probability density function (1), and  (7), the Bayes estimator is (ii) The Bayes estimator under the squared log error loss function (9) is (2 ) Proof. (i) Form Equation (14), it is obviously concluded that the posterior distribution of the parameter θ is Gamma distribution (2 1, ) [ | ] , 2 Thus, the Bayes estimator under the weighted square error loss function (7) is derived as For the case (ii): By using (14), Then the Bayes estimator under the squared log error loss function (9) is come out to be  (12) and (17), the Bayes estimator under the entropy loss function (11) is given by

Minimax Estimation of ЭРланга Distribution
This section will derive the minimax extimators of Э Рланга Distribution by using Lehmann's Theorem, which depends on specific prior distribution and loss functions of a Bayesian method. The Lehmann's Theorem is stated as follows: Lemma 1 Let Proof. To use Lehmann's Theorem for the proof of the results. We need calculate the risk function of Bayes estimators and prove these risk functions are constants.
For the case (i), we can derive the risk function of the Bayes estimator ˆB S δ under the weighted square error loss function (7) as follows: Consequently, , then we can prove that ~(2 ,1) The derivative of ( ) n Ψ is

Performances of Bayes Estimators
To illustrate the performance of these Bayes estimators, squared error loss function 2 ( , ) ( )      From Figure 1 to Figure 4, we know that no of these estimators is uniformly better that other estimators. Then in practice, we recommend to select the estimator according to the prior parameter value d when assuming the quasi-prior as the prior distribution.

Conclusion
This paper derived Bayes estimators of the parameter of Э Рланга distribution under weighted squared error loss, squared log error loss and entropy loss functions. Mote Carlo simulations show that the risk functions of these estimators, defined under squared error loss function, are all decrease as sample size n increases. The risk functions more and more close to each other aehen the sample size n is large, such as n>50.