Discrete Dynamics in Nature and Society

Volume 2015 (2015), Article ID 760768, 5 pages

http://dx.doi.org/10.1155/2015/760768

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

^{1}School of Economics and Statistics, Guangzhou University, Guangzhou 510006, China^{2}College of Science, Guangdong Ocean University, Zhanjiang 524088, China^{3}Department of Mathematics, Shangrao Normal University, Shangrao 334001, China^{4}College of Economics, Jinan University, Guangzhou 510632, China

Received 13 January 2015; Revised 3 May 2015; Accepted 5 May 2015

Academic Editor: Chris Goodrich

Copyright © 2015 Naiyi Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 bywhere

The marginal density function of is shown by

Applying (6), we get

Since , we get

Thus, we havewhere 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 whereUsing Taylor expansion, we get Because is continuous in and condition (C1), it follows thatfurther, we have It is easy to see thatWhen and , we get Substituting (24) and (26) into (20), proof of (I) is finished.*Proof of (II)*. Similarly to (20), we can show thatBy Taylor expansion, we obtain where , due to condition (C1) and , we have Therefore, taking , we getBy (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*

*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, thenIf Theorem 3 holds, we only need to prove .

Applying Markov’s and Jensen’s inequations, we haveAgain using Lemma 1(I), for fixed and , we getSubstituting (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 haveBy 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*

*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*

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

*Acknowledgments*

*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).*

*References*

*References*

- G. A. McIntyre, “A method for unbiased selective sampling, using ranked sets,”
*Australian Journal of Agricultural Research*, vol. 3, no. 4, pp. 385–390, 1952. View at Publisher · View at Google Scholar - Z. H. Chen, Z. D. Bai, and B. K. Sinha,
*Ranked Set Sampling: Theory and Applications*, Springer, 2003. - H. Robbins, “An empirical Bayes approach to statistics,” in
*Proceedings of the 3rd Berkeley Symposium on Mathematical Statistics and Probability*, vol. 1, pp. 157–163, University of California Press, Berkeley, Calif, USA, August 1955. - H. Robbins, “The empirical Bayes approach to statistical decision problems,”
*Annals of Mathematical Statistics*, vol. 35, pp. 1–20, 1964. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. J. Karunamuni, J. Li, and J. Wu, “Robust empirical Bayes tests for continuous distributions,”
*Journal of Statistical Planning and Inference*, vol. 140, no. 1, pp. 268–282, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - X. Xu and D. Zhou, “Empirical Bayes predictive densities for high-dimensional normal models,”
*Journal of Multivariate Analysis*, vol. 102, no. 10, pp. 1417–1428, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - X. Li, Y. Shi, J. Wei, and J. Chai, “Empirical Bayes estimators of reliability performances using LINEX loss under progressively type-II censored samples,”
*Mathematics and Computers in Simulation*, vol. 73, no. 5, pp. 320–326, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - J. Johns and J. van Ryzin, “Convergence rates for empirical Bayes two-action problems. 2. Continuous case,”
*The Annals of Mathematical Statistics*, vol. 43, no. 3, pp. 934–947, 1972. View at Publisher · View at Google Scholar · View at MathSciNet - L. Wang and R. S. Singh, “Linear Bayes estimator for the two-parameter exponential family under type II censoring,”
*Computational Statistics & Data Analysis*, vol. 71, pp. 633–642, 2014. View at Publisher · View at Google Scholar · View at Scopus

*
*