Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2015, Article ID 760768, 5 pages http://dx.doi.org/10.1155/2015/760768
Research Article Empirical Bayes Inference for the Parameter of Power Distribution Based on Ranked Set Sampling Naiyi Li,1,2 Yuan Li,1 Yongming Li,3 and Yang Liu4 1
School of Economics and Statistics, Guangzhou University, Guangzhou 510006, China 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 2
Correspondence should be addressed to Naiyi Li;
[email protected] 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. 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 𝑋 have a conditional density function for given 𝜃: 𝑓 (𝑥 | 𝜃) = (𝜃𝑥)−1 exp {𝜃−1 ln 𝑥} ,
(1)
where 𝜃 is an unknown parameter, Ω = {𝑥 | 𝑥 > 1} is sample space, and Θ = {𝜃 | 𝜃 > 0} 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: 𝐻0 : 𝜃 ≤ 𝜃0 ⇐⇒ 𝐻1 : 𝜃 > 𝜃0 ,
(2)
where 𝜃0 is a given positive constant. To construct test function, we take loss function: 0, 𝜃 ≤ 𝜃0 { { 𝐿 0 (𝜃, 𝑑0 ) = { {𝑎 [1 − ( 𝜃0 )] , 𝜃 > 𝜃 ; 0 𝜃 { 𝜃 {𝑎 [( 0 ) − 1] , 𝜃 ≤ 𝜃0 𝐿 1 (𝜃, 𝑑1 ) = { 𝜃 0, 𝜃 > 𝜃0 , {
(3)
where 𝑎 > 0, 𝑑 = {𝑑0 , 𝑑1 } is action space, respectively, and 𝑑0 and 𝑑1 imply acceptance and rejection of 𝐻0 . Suppose that the prior distribution 𝐺(𝜃) of parameter 𝜃 is unknown, we can get randomized decision function: 𝛿 (𝑥) = 𝑃 (accept 𝐻0 | 𝑋 = 𝑥) .
(4)
2
Discrete Dynamics in Nature and Society where 𝑓𝐺(1) (𝑥) is derivative of 𝑓𝐺(𝑥), 𝑢(𝑥) = 1 − 𝜃0 , and V(𝑥) = −𝜃0 𝑥. By (5), Bayes test function is obtained as follows:
Then, the risk function of 𝛿(𝑥) is given by 𝑅 (𝛿 (𝑥) , 𝐺 (𝜃)) = ∫ ∫ [𝐿 0 (𝜃, 𝑑0 ) 𝑓 (𝑥 | 𝜃) 𝛿 (𝑥) Θ Ω
+ 𝐿 1 (𝜃, 𝑑1 ) 𝑓 (𝑥 | 𝜃) (1 − 𝛿 (𝑥))] 𝑑𝑥 𝑑𝐺 (𝜃)
(5)
= 𝑎 ∫ 𝛽 (𝑥) 𝛿 (𝑥) 𝑑𝑥 + 𝐶𝐺, Ω
{1, if 𝛽 (𝑥) ≤ 0, 𝛿𝐺 (𝑥) = { 0, if 𝛽 (𝑥) > 0. { Hence, we can get minimum Bayes risk:
(11)
𝑅 (𝐺) = inf 𝑅 (𝛿, 𝐺) = 𝑅 (𝛿𝐺, 𝐺)
where
𝛿
(12)
𝐶𝐺 = ∫ 𝐿 1 (𝜃, 𝑑1 ) 𝑑𝐺 (𝜃) , Θ
𝜃 𝛽 (𝑥) = ∫ [1 − ( 0 )] 𝑓 (𝑥 | 𝜃) 𝑑𝐺 (𝜃) . 𝜃 Θ
(6)
𝑓𝐺 (𝑥) = ∫ 𝑓 (𝑥 | 𝜃) 𝑑𝐺 (𝜃) (7)
= ∫ (𝜃𝑥)−1 exp {𝜃−1 ln 𝑥} 𝑑𝐺 (𝜃) . Θ
Applying (6), we get 𝛽 (𝑥) = ∫ (1 − Θ
𝜃0 ) 𝑓 (𝑥 | 𝜃) 𝑑𝐺 (𝜃) 𝜃
= ∫ (1 − 𝜃0 + 𝜃0 − Θ
𝜃0 ) 𝑓 (𝑥 | 𝜃) 𝑑𝐺 (𝜃) 𝜃
= ∫ (1 − 𝜃0 ) 𝑓 (𝑥 | 𝜃) 𝑑𝐺 (𝜃) Θ
+ ∫ (𝜃0 − Θ
𝜃0 ) 𝑓 (𝑥 | 𝜃) 𝑑𝐺 (𝜃) 𝜃
= (1 − 𝜃0 ) 𝑓𝐺 (𝑥) − 𝜃0 𝑥
(8)
1 ⋅ ∫ (− ) 𝑓 (𝑥 | 𝜃) 𝑑𝐺 (𝜃) − 𝜃0 𝑥 𝑥 Θ
Since 𝑓𝐺(𝑥) = ∫Θ (𝜃𝑥)−1 exp{𝜃−1 ln 𝑥}𝑑𝐺(𝜃), we get
1 + ∫ 𝑓 (𝑥 | 𝜃) 𝑑𝐺 (𝜃) . 𝜃𝑥 Θ
(9)
= 𝑢 (𝑥) 𝑓𝐺 (𝑥) + V (𝑥) 𝑓𝐺(1) (𝑥) ,
1
𝑘 𝑚
∑∑ 𝐾𝑟 (
𝑥 − 𝑋(𝑖)𝑗
𝑚𝑘ℎ𝑛(1+𝑟) 𝑖=1𝑗=1
ℎ𝑛
),
(13)
(14)
where ℎ𝑛 is a positive and smoothing bandwidth and lim𝑛 → ∞ ℎ𝑛 = 0. Denote by 𝑓𝑛(0) (𝑥) = 𝑓𝑛 (𝑥), 𝑓𝑛(𝑟) (𝑥) the 𝑟th order derivative of 𝑓𝑛 (𝑥), for 𝑟 = 0, 1. The estimator of 𝛽(𝑥) is obtained by 𝛽𝑛 (𝑥) = 𝑢 (𝑥) 𝑓𝑛 (𝑥) + V (𝑥) 𝑓𝑛(1) (𝑥) .
Thus, we have 𝛽 (𝑥) = (1 − 𝜃0 ) 𝑓𝐺 (𝑥) − 𝜃0 𝑥 ⋅ 𝑓𝐺(1) (𝑥)
1 1 𝑡 ∫ 𝑦 𝐾𝑟 (𝑦) 𝑑𝑦 𝑡! 0
𝑓𝑛(𝑟) (𝑥) =
1 𝑓 (𝑥 | 𝜃) 𝑑𝐺 (𝜃) . 𝜃𝑥
1 𝑓𝐺(1) (𝑥) = ∫ (− ) 𝑓 (𝑥 | 𝜃) 𝑑𝐺 (𝜃) 𝑥 Θ
A balanced RSS procedure can be described by Chen et al. [2]. Under the following conditions, we need to construct EB test function. Let 𝑋(1)1 , 𝑋(1)2 , . . . , 𝑋(1)𝑚 , 𝑋(2)1 , 𝑋(2)2 , . . . , 𝑋(2)𝑚 , . . . , 𝑋(𝑘)1 , 𝑋(𝑘)2 , . . . , 𝑋(𝑘)𝑚 be a balanced ranked set sample from population which has the common marginal density function 𝑓𝐺(𝑥). We assume perfect ranking. Denote by 𝑋(1)1 ,𝑋(1)2 , . . . , 𝑋(1)𝑚 , 𝑋(2)1 , 𝑋(2)2 , . . . , 𝑋(2)𝑚 , . . . , 𝑋(𝑘)1 , 𝑋(𝑘)2 , . . . , 𝑋(𝑘)𝑚 the historical samples, and 𝑋 is present sample. Assume 𝑓𝐺(𝑥) ∈ 𝐶𝑠,𝛼 , 𝑥 ∈ 𝑅1 , where 𝐶𝑠,𝛼 = {𝑔(𝑥) | 𝑔(𝑥) is a probability density function and has continuous 𝑠th order derivative 𝑔(𝑠) (𝑥) with |𝑔(𝑠) (𝑥)| ≤ 𝛼, 𝑠 ≥ 3, 𝛼 > 0}. First construct estimator of 𝛽(𝑥). Let 𝐾𝑟 (𝑥) be a Borel measurable bounded function vanishing off (0, 1) such that
𝑡 {(−1) , when 𝑡 = 𝑟, ={ 0, when 𝑡 ≠ 𝑟, 𝑡 = 0, 1, 2, . . . , 𝑠 − 1. { Kernel estimator of 𝑓𝐺(𝑥) is defined by
= (1 − 𝜃0 ) 𝑓𝐺 (𝑥) − 𝜃0 𝑥
Θ
2. Construction of EB Test Based on Ranked Set Sampling
(𝐶1) :
1 1 ⋅ ∫ (−1 + ) 𝑓 (𝑥 | 𝜃) 𝑑𝐺 (𝜃) 𝜃 𝑥 Θ
⋅∫
Ω
If the prior distribution of 𝐺(𝜃) is known and 𝛿(𝑥) = 𝛿𝐺(𝑥), 𝑅(𝐺) is achieved. While 𝐺(𝜃) is unknown, we cannot make use of 𝛿𝐺(𝑥) and need to introduce EB method.
The marginal density function of 𝑋 is shown by Θ
= 𝑎 ∫ 𝛽 (𝑥) 𝛿𝐺 (𝑥) 𝑑𝑥 + 𝐶𝐺.
(15)
Hence, EB test function is defined by (10)
{1, 𝛽𝑛 (𝑥) ≤ 0, 𝛿𝑛 (𝑥) = { 0, 𝛽𝑛 (𝑥) > 0. {
(16)
Discrete Dynamics in Nature and Society
3
Let 𝐸 stand for mathematical expectation with respect to the joint distribution of 𝑋(1)1 , 𝑋(1)2 , . . . , 𝑋(1)𝑚 , 𝑋(2)1 , 𝑋(2)2 , . . . , 𝑋(2)𝑚 , . . . , 𝑋(𝑘)1 , 𝑋(𝑘)2 , . . . , 𝑋(𝑘)𝑚 . Hence, we get the overall Bayes risk of 𝛿𝑛 (𝑥): 𝑅 (𝛿𝑛 (𝑥) , 𝐺) = 𝑎 ∫ 𝛽 (𝑥) 𝐸 [𝛿𝑛 (𝑥)] 𝑑𝑥 + 𝐶𝐺. Ω
(17)
If lim𝑛 → ∞ 𝑅(𝛿𝑛 , 𝐺) = 𝑅(𝛿𝐺, 𝐺), {𝛿𝑛 (𝑥)} is called asymptotical optimality of EB test function. If 𝑅(𝛿𝑛 , 𝐺) − 𝑅(𝛿𝐺, 𝐺) = 𝑂(𝑛−𝑞 ), where 𝑞 > 0, 𝑂(𝑛−𝑞 ) is asymptotically optimal convergence rate of EB test function {𝛿𝑛 (𝑥)}. Before proving the theorems, we need the following lemmas. Let 𝑐, 𝑐1 , 𝑐2 , 𝑐3 be different constants in different cases even in the same expression. Lemma 1. Let 𝑋(1)1 , 𝑋(1)2 , . . . , 𝑋(1)𝑚 , 𝑋(2)1 , 𝑋(2)2 , . . . , 𝑋(2)𝑚 , . . . , 𝑋(𝑘)1 , 𝑋(𝑘)2 , . . . , 𝑋(𝑘)𝑚 be balanced ranked set samples. Suppose that (C1) holds, ∀𝑥 ∈ Ω. 𝑓𝐺(𝑟) (𝑥)
(I) When is continuous function, lim𝑛 → ∞ ℎ𝑛 = 0, and lim𝑛 → ∞ 𝑛ℎ𝑛2𝑟+1 = ∞, one has 2 (18) lim 𝐸 𝑓(𝑟) (𝑥) − 𝑓𝐺(𝑟) (𝑥) = 0. 𝑛 → ∞ 𝑛 (II) When 𝑓𝐺(𝑥) ∈ 𝐶𝑠,𝑎 , putting ℎ𝑛 = 𝑛−1/(2+𝑠) , for 0 < 𝜆 ≤ 1, one has 2𝜆 (19) 𝐸 𝑓𝑛(𝑟) (𝑥) − 𝑓𝐺(𝑟) (𝑥) ≤ 𝑐 ⋅ 𝑛−𝜆(𝑠−2𝑟+1)/(2+𝑠) .
Proof of (I). Using 𝐶𝑟 inequation, we obtain 2 𝐸 𝑓𝑛(𝑟) (𝑥) − 𝑓𝐺(𝑟) (𝑥) 2 ≤ 2 𝐸𝑓𝑛(𝑟) (𝑥) − 𝑓𝐺(𝑟) (𝑥) + 2 Var (𝑓𝑛(𝑟) (𝑥)) :=
(𝑥) =
𝑘 𝑚
∑∑ 𝑛−1 ℎ𝑛−(𝑟+1) 𝐸 [𝐾𝑟 𝑖=1 𝑗=1
∞
0
(
𝑥 − 𝑋(𝑖)𝑗
𝑥 − 𝑋(𝑖)𝑗 ℎ𝑛
𝑖=1 𝑗=1
≤
𝑘 𝑚 2𝑛−2 ℎ𝑛−2(𝑟+1) ∑ ∑𝐸 [𝐾𝑟 𝑖=1 𝑗=1
(
𝑥 − 𝑋(𝑖)𝑗 ℎ𝑛
)] (25)
2
)]
−1
≤ 𝑐 ⋅ (𝑛ℎ𝑛2𝑟+1 ) .
(26)
𝑛→∞
Substituting (24) and (26) into (20), proof of (I) is finished. (20)
Proof of (II). Similarly to (20), we can show that
ℎ𝑛
≤ 2 [𝐸𝑓𝑛(𝑟) (𝑥) − 𝑓𝐺(𝑟) (𝑥)] + 2 [Var 𝑓𝑛(𝑟) (𝑥)]
)]
(21)
𝑓𝐺 (𝑥 − ℎ𝑛 V) = 𝑓𝐺 (𝑥) + +
1
∫ 𝐾𝑟 (𝑢) 𝑓𝐺 (𝑥 − ℎ𝑛 𝑢) 𝑑𝑢. 0
(22)
𝑓𝐺 (𝑥) (−ℎ𝑛 V) 1!
𝑓𝐺 (𝑥) 2 (−ℎ𝑛 V) + ⋅ ⋅ ⋅ 2!
(28)
𝑓𝐺(𝑠) (𝑥 − 𝜉ℎ𝑛 V) 𝑠 (−ℎ𝑛 V) , 𝑠! where 0 < 𝜉 < 1, due to condition (C1) and 𝑓𝐺(𝑥) ∈ 𝐶𝑠,𝛼 , we have (𝑟) 𝐸𝑓𝑛 (𝑥) − 𝑓𝐺(𝑟) (𝑥) (𝑠) (29) 1 𝑠−𝑟 𝑠 𝑓𝐺 (𝑥 − 𝜉ℎ𝑛 V) 𝑠−𝑟 ≤ ∫ 𝐾𝑟 (V) ℎ𝑛 V 𝑑V ≤ 𝑐 ⋅ ℎ𝑛 . 𝑠! 0 +
𝑓𝐺 (𝑥 − ℎ𝑛 𝑢) − 𝑓𝐺 (𝑢)
𝑓𝐺(𝑠) (𝑥 − 𝜉ℎ𝑛 𝑢) 𝑠 (−ℎ𝑛 𝑢) . 𝑠!
(27)
By Taylor expansion, we obtain
𝑥−𝑦 ) 𝑓𝐺 (𝑦) 𝑑𝑦 ℎ𝑛
𝑓 (𝑥) 𝑓𝐺 (𝑥) 2 (−ℎ𝑛 𝑢) + 𝐺 (−ℎ𝑛 𝑢) + ⋅ ⋅ ⋅ 1! 2!
𝜆
:= 2 (𝐵12𝜆 + 𝐵2𝜆 ) .
Using Taylor expansion, we get
+
𝑘 𝑚
= 2∑ ∑𝑛−2 ℎ𝑛−2(𝑟+1) Var [𝐾𝑟 (
2𝜆
𝑥−𝑋 )] ℎ𝑛
= ℎ𝑛−(1+𝑟) ∫ 𝐾𝑟 (
=
𝐴 2 = 2 Var [𝑓𝑛(𝑟) (𝑥)]
2𝜆 𝐸 𝑓𝑛(𝑟) (𝑥) − 𝑓𝐺(𝑟) (𝑥)
= ℎ𝑛−(1+𝑟) 𝐸 [𝐾𝑟 (
=
(24)
It is easy to see that
lim 𝐴 2 = lim Var (𝑓𝐺(𝑟) (𝑥)) = 0.
+ 𝐴 2) ,
ℎ𝑛−𝑟
2 lim 𝐴21 = lim 𝐸𝑓𝑛(𝑟) (𝑥) − 𝑓𝐺(𝑟) (𝑥) = 0. 𝑛→∞
𝑛→∞
𝑛→∞
where 𝐸𝑓𝑛(𝑟)
further, we have
When ℎ𝑛 → 0 and 𝑛ℎ𝑛2𝑟+1 → ∞, we get
Proof. Consider the following.
2 (𝐴21
Because 𝑓𝐺(𝑟) (𝑥) is continuous in 𝑥 and condition (C1), it follows that 0 ≤ lim 𝐸𝑓𝑛(𝑟) (𝑥) − 𝑓𝐺(𝑟) (𝑥) 𝑛→∞ 1 1 = lim 𝑟 ∫ 𝐾𝑟 (𝑢) 𝑓𝐺 (𝑥 − ℎ𝑛 𝑢) 𝑑𝑢 − 𝑓𝐺(𝑟) (𝑥) 𝑛→∞ ℎ 0 𝑛 (23) 1 1 𝑟 ≤ ∫ 𝑢 𝐾𝑟 (𝑢) 𝑟! 0 ⋅ lim 𝑓𝐺(𝑟) (𝑥 − 𝜉ℎ𝑛 𝑢) − 𝑓𝐺(𝑟) (𝑥) 𝑑𝑢 = 0; 𝑛→∞
4
Discrete Dynamics in Nature and Society Therefore, taking ℎ𝑛 = 𝑛−1/(2+𝑠) , we get 2𝜆 𝐵12𝜆 = 𝐸𝑓𝑛(𝑟) (𝑥) − 𝑓𝐺(𝑟) (𝑥) ≤ 𝑐 ⋅ 𝑛−2𝜆(𝑠−𝑟)/(𝑠+2) .
(30)
By (26), choosing ℎ𝑛 = 𝑛−1/(2+𝑠) , we can get −1 𝜆
𝐵2𝜆 ≤ [𝑐1 (𝑛ℎ𝑛2𝑟+1 ) ] ≤ 𝑐 ⋅ 𝑛−𝜆(𝑠−2𝑟+1)/(2+𝑠) .
(31)
Substituting (30) and (31) into (27), proof of (II) is finished.
If Theorem 3 holds, we only need to prove lim𝑛 → ∞ 𝑀𝑛 (𝑥) = 0 a.s. 𝑥. Applying Markov’s and Jensen’s inequations, we have 𝑀𝑛 (𝑥) ≤ 𝐸 𝛽𝑛 (𝑥) − 𝛽 (𝑥) ≤ |𝑢 (𝑥)| 𝐸 𝑓𝑛 (𝑥) − 𝑓𝐺 (𝑥) + |V (𝑥)| 𝐸 𝑓𝑛(1) (𝑥) − 𝑓𝐺(1) (𝑥) (37) 2 1/2 ≤ |𝑢 (𝑥)| [𝐸 𝑓𝑛 (𝑥) − 𝑓𝐺 (𝑥) ] 2 + |V (𝑥)| [𝐸 𝑓𝑛(1) (𝑥) − 𝑓𝐺(1) (𝑥) ]
Lemma 2 (see [8]). 𝑅(𝛿𝐺, 𝐺) and 𝑅(𝛿𝑛 , 𝐺) are defined by (12) and (17); then
.
Again using Lemma 1(I), for fixed 𝑥 ∈ Ω and 𝑟 = 0, 1, we
0 ≤ 𝑅 (𝛿𝑛 , 𝐺) − 𝑅 (𝛿𝐺, 𝐺) ≤ 𝑎 ∫ 𝛽 (𝑥) 𝑃 (𝛽𝑛 (𝑥) − 𝛽 (𝑥) ≥ 𝛽 (𝑥)) 𝑑𝑥. Ω
1/2
(32)
get 0 ≤ lim 𝑀𝑛 (𝑥) 𝑛→∞
2 1/2 ≤ |𝑢 (𝑥)| [ lim 𝐸 𝑓𝑛 (𝑥) − 𝑓𝐺 (𝑥) ] 𝑛→∞
3. Asymptotic Optimality and Convergence Rates Based on Ranked Set Sampling Theorem 3. Assume (C1) and the following regularity conditions hold:
(38)
2 1/2 + |V (𝑥)| [ lim 𝐸 𝑓𝑛(1) (𝑥) − 𝑓𝐺(1) (𝑥) ] = 0. 𝑛→∞ Substituting (38) into (36), proof of Theorem 3 is finished.
(i) lim𝑛 → ∞ 𝑛ℎ𝑛3 = ∞; (ii) ∫Θ 𝜃−2 𝑑𝐺(𝜃) < +∞; (iii)
𝑓𝐺(1) (𝑥)
is continuous function, and one has lim 𝑅 (𝛿𝑛 , 𝐺) = 𝑅 (𝛿𝐺, 𝐺) .
𝑛→∞
(33)
𝑅 (𝛿𝑛 , 𝐺) − 𝑅 (𝛿𝐺, 𝐺) = 𝑂 (𝑛−𝜆(𝑠−1)/2(𝑠+1) ) ,
Proof of Theorem 3. Applying Lemma 2, we can get
𝑤ℎ𝑒𝑟𝑒 𝑠 ≥ 2.
0 ≤ 𝑅 (𝛿𝑛 , 𝐺) − 𝑅 (𝛿𝐺, 𝐺) ≤ 𝑎 ∫ 𝛽 (𝑥) 𝑝 (𝛽𝑛 (𝑥) − 𝛽 (𝑥) ≥ 𝛽 (𝑥)) 𝑑𝑥. Ω
(34)
Ω Θ
=
1 + 𝜃02
Proof of Theorem 4. Applying Lemma 2 and Markov’s inequations, we have
1−𝜆 𝜆 ≤ ∫ 𝛽 (𝑥) 𝐸 𝛽𝑛 (𝑥) − 𝛽 (𝑥) 𝑑𝑥 Ω 1−𝜆 ≤ 𝑐1 ∫ 𝛽 (𝑥) |𝑢 (𝑥)| 𝐸 𝑓𝑛 (𝑥) − 𝑓𝐺 (𝑥) 𝑑𝑥 Ω
∫ 𝛽 (𝑥) 𝑑𝑥 Ω 𝜃0 2 ) ] 𝑓 (𝑥 | 𝜃) 𝑑𝐺 (𝜃) 𝑑𝑥 𝜃
(35)
∫ ∫ 𝜃 𝑓 (𝑥 | 𝜃) 𝑑𝐺 (𝜃) 𝑑𝑥
= 𝐷𝑛 + 𝐸𝑛 . By Lemma 1(II) and condition (C2), we get
Ω Θ
1−𝜆 𝐷𝑛 ≤ 𝑐1 𝑛−𝜆𝑠/(2𝑠+2) ∫ 𝛽 (𝑥) |𝑢 (𝑥)|𝜆 𝑑𝑥 Ω
= 1 + 𝜃02 ∫ 𝜃−2 𝑑𝐺 (𝜃) < + ∞. Θ
≤ 𝑐3 𝑛−𝜆𝑠/(2𝑠+2) ,
By domain convergence theorem, then 0 ≤ lim 𝑅 (𝛿𝑛 , 𝐺) − 𝑅 (𝛿𝐺, 𝐺) 𝑛→∞
Ω 𝑛→∞
(40)
1−𝜆 + 𝑐2 ∫ 𝛽 (𝑥) |V (𝑥)| 𝐸 𝑓𝑛(1) (𝑥) − 𝑓𝐺(1) (𝑥) 𝑑𝑥 Ω
−2
≤ ∫ [ lim 𝑀𝑛 (𝑥)] 𝑑𝑥.
(39)
0 ≤ 𝑅 (𝛿𝑛 , 𝐺) − 𝑅 (𝛿𝐺, 𝐺)
Put 𝑀𝑛 (𝑥) = |𝛽(𝑥)|𝑝(|𝛽𝑛 (𝑥) − 𝛽(𝑥)| ≥ |𝛽(𝑥)|). It is easy to see that 𝑀𝑛 (𝑥) ≤ |𝛽(𝑥)|. Again using (10) and Fubini theorem, we obtain
= ∫ ∫ [1 − (
Theorem 4. Assume (C1) and the following regularity conditions hold. (C2): consider ∫Ω 𝑥−𝑚𝜆 |𝛽(𝑥)|1−𝜆 𝑑𝑥 < +∞, where 0 < 𝜆 ≤ 1, 𝑚 = 0, 1. When ℎ𝑛 = 𝑛−1/(2+2𝑠) , where 𝑠 ≥ 2, one can obtain
(36)
𝐸𝑛 ≤ 𝑐2 𝑛
−𝜆(𝑠−1)/(2𝑠+2)
1−𝜆 ∫ 𝛽 (𝑥) |V (𝑥)|𝜆 (𝑥) 𝑑𝑥 Ω
≤ 𝑐4 𝑛−𝜆(𝑠−1)/(2𝑠+2) .
(41)
Discrete Dynamics in Nature and Society Substituting (41) into (40), we can obtain 𝑅(𝛿𝑛 , 𝐺) − 𝑅(𝛿𝐺, 𝐺) = 𝑂(𝑛−𝜆(𝑠−1)/2(𝑠+1) ). Proof of Theorem 4 is finished. Remark 5. When 𝜆 → 1 and 𝑠 → ∞, 𝑂(𝑛−𝜆(𝑠−1)/2(𝑠+1) ) nears 𝑂(𝑛−1/2 ).
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).
References [1] 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. [2] Z. H. Chen, Z. D. Bai, and B. K. Sinha, Ranked Set Sampling: Theory and Applications, Springer, 2003. [3] 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. [4] H. Robbins, “The empirical Bayes approach to statistical decision problems,” Annals of Mathematical Statistics, vol. 35, pp. 1– 20, 1964. [5] 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. [6] 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. [7] 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. [8] 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.
5 [9] L. Wang and R. S. Singh, “Linear Bayes estimator for the twoparameter exponential family under type II censoring,” Computational Statistics & Data Analysis, vol. 71, pp. 633–642, 2014.
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