Empirical Bayes One-side Test for Inverse Exponential Model Based on Negative Associate Samples

By using the kernel-type density estimation and empirical distribution function in the case of identically distributed and negatively associated samples, the empirical Bayes one-sided test rules for the parameter of inverse exponential distribution are constructed based on negative associate sample under weighted linear loss function, and the asymptotically optimal property is obtained . It is shown that the convergence rates of the proposed empirical Bayes test rules can arbitrarily close to 1/2 ( ) − Ο n under suitable conditions.


Introduction
Empirical Bayes method, first proposed by Robbins [1], is a very important method in statistical inference field, and it has many applications in the field of reliability, lifetime prediction, medical research, ect. [2][3][4][5][6]. Empirical Bayes hypothesis test problem is a hot topic, which attracts many scholars' attention.
For example, Karunamuni [7] discussed the optimal convergence rate of empirical Bayes tests for uniform distributions. Xu et al. [8] investigated the convergence rates of empirical Bayes tests of parameters in one-side truncated distribution families under an asymmetric loss function. Qian and Wei [9] studied the two-sided empirical Bayes test for a class of scale exponential model under a weighed loss function. Chen et al. [10] studied the two-sided empirical Bayes test problem for the continuous one-parameter exponential model with contaminated data by a kernel probability density method. Under Type-II censored samples. Wang et al. [11] studied the empirical Bayes test problem for two-parameter exponential distribution. However, in many situations, such as penetration theory, reliability analysis and multivariable analysis, random samples are probably not independent identical distribution but negatively associated (NA) samples [12,13]. For instance, Wei [14] developed a empirical Bayes test of parameter for exponential distribution in the case of NA samples using the linear weighted loss functions. Shi et al. [15] developed a two-sided empirical Bayes test for the parameter of the truncated distribution family in the case of NA samples. Wang and Shi [16] studied the Bayes test problem for a special exponential family under NA samples.
The definition of the NA sequence was first introduced by Esary in [17] as follows: Definition 1. A sequence of random variables 1 2 , ,..., n X X X is said to be a NA sample, if for every pair of disjoint non-empty subsets 1 A and 2 A of set {1, 2,..., } n , , ,..., n X X X is negatively associated.
Assume X is a random variable drawn form inverse exponential distribution with the probability density function is The inverse exponential distribution is one of the most important lifetime distributions [18], and the applications in life test, reliability and statistical inference attract many authors' attention. For example, Prakash [19] discussed the Bayes estimation of the inverse exponential model under symmetric and asymmetric loss functions. Rao [20] studied the reliability estimation for the stress-strength model, which the stress and strength are both distributed with inverse exponential distribution. Singh et al. [21] developed a twostage group acceptance sampling plan for a generalized inverse exponential distribution under truncated life test.
In this paper, we will first construct the empirical Bayes test function for testing Eq. (2) in inverse exponential model. Then, we will prove that the proposed test is asymptotically optimal and its convergence rate will also be derived under certain suitable conditions. The remains of this paper is organized as follows: Section 2 will establish a Bayes decision rule for one-side hypothesis test problem. Then Section 3 will put forward a new empirical Bayes decision rule combining the kernel-type density function with empirical distribution function. Finally, Section 4 will give the conclusion of this paper.

Bayes Test
Given the inverse exponential distribution with probability density function (1), the hypothesis needed to be tested is Here 0 θ is a given constant. This section will construct a new Bayes test rule for hypothesis test problem (2) under the following weighted linear loss functions: The weighted linear loss function has many advantages: it is invariant and makes the expression of Bayes test function more concise and makes the empirical Bayes test function easy to construct.
Suppose that X has conditional probability density in Eq. (1) and parameter θ has the unknown prior distribution function ( ) G θ , ( ) g θ is the corresponding prior probability density function. Then the marginal probability density function of X is: According to equation (5), we can get A test is said to be a Bayes test with respect to ( ) It is easy to show that ( ) G x δ is a Bayes test with respect to ( ) The Bayes risk function is:

Empirical Bayes Test Function
This section we will construct the empirical Bayes test function. Suppose that 1 2 , ,..., n X X X ,  I X x n (13) Therefore, the estimator of ( ) x Finally, the empirical Bayes test function could be defined as follows: Assume n E represents the mean value of the joint distribution of 1 2 , ,..., n X X X , hence the comprehensive risk Then we say the convergence rate of the test function By Definitions 1 and 2, we conclude that the optimal evaluation of the empirical Bayes test functions depends on the degree of its risk approximation Bayes risk.
In this paper, we let 1 2 , , ,... c c c denote different positive constants in different cases even in the same expression. In order to prove Theorems 1 and 2, which will be illustrated in Section 4, we need to present the following lemmas:   (8) and equation (13) separately, 1 2 , ,... X X are identically distributed and weakly stationary NA sequence, for Proof. According to equation (14), we have