Empirical Bayes Inference for the Parameter of Power Distribution Based on Ranked Set Sampling

Abstract

This research is based on ranked set sampling. Through the analysis and proof, the empirical Bayes test rule and asymptotical property for the parameter of power distribution are obtained.

1. Introduction

Ranked set sampling (RSS) is now regarded as an effective tool in statistical inference and important alternative to simple random sampling. RSS was first applied in agriculture [1]. In recent years, it has been applied more and more in areas such as environment and ecology. It is also a potential for the method to be successfully applied in industrial statistics and sociology, for which readers can refer to the monograph by Chen et al. [2].

Empirical Bayes (EB) approach was originally proposed by Robbins [3, 4], soon after it had been studied in the literature [5–9]. Up to now, EB methods are commonly based on simple random sampling (SRS). As a statistical procedure based on the RSS would perform better than its counterpart based on simple random sampling (SRS), a natural idea is to develop EB methods based on RSS. We will construct empirical Bayes test rule for the parameter of power distribution based on RSS.

Let have a conditional density function for given : where is an unknown parameter, is sample space, and is parameter space.

In the paper, the main motivation of studying the density of (1) is as follows: (a) it is widely used in the fields of reliability and economy and so on; (b) it is a common distribution in reliability distribution; (c) on the basis of studying it, we intend to further consider empirical Bayes inference for some other distributions based on ranked set sampling. In the paper, we study the following test problem: where is a given positive constant.

To construct test function, we take loss function: where , is action space, respectively, and and imply acceptance and rejection of .

Suppose that the prior distribution of parameter is unknown, we can get randomized decision function:

Then, the risk function of is given by where

The marginal density function of is shown by

Applying (6), we get

Since , we get

Thus, we have where is derivative of , , and .

By (5), Bayes test function is obtained as follows:

Hence, we can get minimum Bayes risk:

If the prior distribution of is known and , is achieved. While is unknown, we cannot make use of and need to introduce EB method.

2. Construction of EB Test Based on Ranked Set Sampling

A balanced RSS procedure can be described by Chen et al. [2]. Under the following conditions, we need to construct EB test function. Let be a balanced ranked set sample from population which has the common marginal density function . We assume perfect ranking. Denote by the historical samples, and is present sample. Assume , , where is a probability density function and has continuous th order derivative   with  . First construct estimator of .

Let be a Borel measurable bounded function vanishing off such that

Kernel estimator of is defined by where is a positive and smoothing bandwidth and .

Denote by , the th order derivative of , for .

The estimator of is obtained by

Hence, EB test function is defined by

Let stand for mathematical expectation with respect to the joint distribution of . Hence, we get the overall Bayes risk of :

If , is called asymptotical optimality of EB test function. If , where , is asymptotically optimal convergence rate of EB test function . Before proving the theorems, we need the following lemmas.

Let , , , be different constants in different cases even in the same expression.

Lemma 1. Let be balanced ranked set samples. Suppose that (C1) holds, . (I)When is continuous function, , and , one has (II)When , putting , for , one has

Proof. Consider the following.
Proof of (I). Using inequation, we obtain where Using Taylor expansion, we get Because is continuous in and condition (C1), it follows that further, we have It is easy to see that When and , we get Substituting (24) and (26) into (20), proof of (I) is finished.
Proof of (II). Similarly to (20), we can show that By Taylor expansion, we obtain where , due to condition (C1) and , we have Therefore, taking , we get By (26), choosing , we can get Substituting (30) and (31) into (27), proof of (II) is finished.

Lemma 2 (see [8]). and are defined by (12) and (17); then

3. Asymptotic Optimality and Convergence Rates Based on Ranked Set Sampling

Theorem 3. Assume (C1) and the following regularity conditions hold: (i); (ii); (iii) is continuous function, and one has

Proof of Theorem 3. Applying Lemma 2, we can get Put .
It is easy to see that .
Again using (10) and Fubini theorem, we obtain By domain convergence theorem, then If Theorem 3 holds, we only need to prove .
Applying Markov’s and Jensen’s inequations, we have Again using Lemma 1(I), for fixed and , we get Substituting (38) into (36), proof of Theorem 3 is finished.

Theorem 4. Assume (C1) and the following regularity conditions hold.
(C2): consider , where , . When , where , one can obtain

Proof of Theorem 4. Applying Lemma 2 and Markov’s inequations, we have By Lemma 1(II) and condition (C2), we get Substituting (41) into (40), we can obtain . Proof of Theorem 4 is finished.

Remark 5. When and , nears .

4. Conclusions

In this paper, we propose the empirical Bayes test rule for the parameter of power distribution. Based on ranked set sampling, the asymptotical optimality and convergence rates of EB test function are obtained. Another extension of this work would consider empirical Bayes inference for some other distributions based on ranked set sampling.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the anonymous referees for constructive suggestions that greatly improved the paper. This work was partially supported by the National Natural Science Foundation of China (11271095 and 11461057) and Natural Science Foundation of Guangdong Ocean University (GDOU2014050217). The work was also supported by the Specialized Research Fund for the Doctoral Program of Higher Education (20124410110002) and National Statistics Projects of China (2012LY178).