Journal of Statistical Theory and Applications

Volume 17, Issue 2, June 2018, Pages 291 - 306

Performance of the Graybill–Deal Estimator via Pitman Closeness Criterion

Authors
Keyu Nie*, Bikas.K. Sinha, A.S. Hedayat
Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago
Received 23 September 2016, Accepted 8 May 2017, Available Online 30 June 2018.
DOI
https://doi.org/10.2991/jsta.2018.17.2.9How to use a DOI?
Keywords
Graybill-Deal Estimator; Pitman Closeness; Meta Analysis
Abstract

Pitman closeness criterion is a coverage probability-based criterion to examine the relative performances of estimators. Usually, the performance of the standard Graybill-Deal estimator of the common mean has been examined with respect to the mean squared error (variance). In this study we examine its performance with respect to the Pitman closeness criterion. Specifically, we compare a p-source based Graybill-Deal estimator against its q-source based competitors for q (< p)-source subsets of p-source data. The key references to this paper are [5] and [7].

Open Access
This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

1. Introduction

We consider the problem of estimation of the common mean shared by several independent normal populations with unknown and most likely unequal variances. Generally, we have p independent sources with distributions N(μ,σi2) , i = 1,2,…, p. Also, let x¯i and si2 be sample mean and sample variance, respectively; and sx¯i2=si2/ni , σx¯i2=σi2/ni , where ni is the sample size available from the ith source, i = 1,2,…, p.

[1] introduced their estimator μˆGD|2 for p = 2 sources, and claimed that μˆGD|2 was preferable to both sample means with respect to the criteria of mean square error, if and only if sample sizes n1 and n2 were moderate enough (≥ 11), which was corrected by [4] as (n1 ≥ 11,n2 ≥ 11), (n1 = 10,n2 ≥ 19) or (n1 ≥ 19,n2 = 10). Subsequently this result was extended by [3] to include p independent sources. They compared the Graybill-Deal estimator (GDE) of combining p sources

μˆGD|p=i=1px¯isx¯i2i=1p1sx¯i2
with sample means x¯i , i = 1,2,…, p, and found that μˆGD|p had a smaller variance than each sample mean under the same condition as [1]; that is, either ni > 10 for i = 1,2,…, p, or ni = 10 for some i, and nj > 18 (i, j = 1,2,…, p for each ji). [8] showed that μˆGD|p dominated any μˆGD|q of q (< p) sub-sources, under the same condition as in [3], with respect to mean square criterion.

Here we shall compare μˆGD|p with μˆGD|q by employing Pitman closeness criterion, which was introduced by [6]: We say estimator μˆ1 is better (Pitman-closer) than μˆ2 for the estimation of the parameter µ if and only if P{|μˆ1μ||μˆ2μ|}1/2 . Pitman-closeness criterion is a coverage probability-based criterion and has nothing to do with the loss function. It provides another angle in the sense of ’one-to-one’ comparison measuring the performance efficiency of GDE.

Using the Pitman-closeness criterion, [5] and [7] established that for p = 2, a necessary and sufficient condition for

P{|μˆGD|2μ||x¯iμ|}1/2,
to hold uniformly in (μ,σx¯12,σx¯22) is that mi = ni − 1 ≥ 4 for each i = 1,2. [7] further established that
P{|μˆGD|pμ||x¯iμ|}1/2,
for all i = 1,2,…, p and uniformly in (μ,σx¯12,σx¯22,,σx¯p2) , if and only if
2E{(j=1,jipσx¯j2Yj2)1/2(j=1,jipσx¯j2Yj)}E{(j=1,jipσx¯j2Yj2)1/2},
holds for all i = 1,2,…, p and uniformly in (σx¯12,σx¯22,,σx¯p2) , where Yj’s are independently distributed as mjχ2(mj) and mj = nj − 1, for j = 1,2,…, p. [7] also gave a sufficient condition by showing that inequality (1.1) holds, for any i = 1,2,…, p, if
18mk1+4(j=1,jipmj)10,
for all k in {1,2,…,i − 1,i + 1,…, p}.

In this paper we show that μˆGD|p is a Pitman-closer estimator than μˆGD|q , the GDE of any q (< p) sub-sources, in the sense of Pitman closeness criteria for all p > 2. Without loss of generality, our results are presented by comparing with initial q sub-sources, although for any other q sub-sources, we will obtain similar results.

We provide several sufficient conditions [including the inequality (1.1) (suitably modified) which holds uniformly in (σx¯12,σx¯22,,σx¯p2) for all q sub-sources]. We also provide a necessary condition towards this.

The remainder of this paper is structured as follows. Section 2 describes all the lemmas and preparation for the main results. Section 3 provides a necessary condition when combining all p-sources is preferred. Section 4 presents the corresponding sufficient condition and several corollaries. Section 5 discusses the sample size requirement. Section 6 concludes the paper.

2. Notations and Lemmas

Before introducing our main results, we first introduce the following notations and lemmas.

We borrow the definition from [7]:

Let

Yi=σx¯i2sx¯i2~miχ2(mi),fori=1,2,,p,
where Yi’s are independently distributed and mi = ni − 1 is the degrees of freedom. Obviously we have E(Yi1)=1 .

Let p = q + r and define:

Uq=(i=1qσx¯i2sx¯i4)12(i=1qsx¯i2)=(i=1qσx¯i2Yi2)12(i=1qσx¯i2Yi),Ur=(i=q+1q+rσx¯i2sx¯i4)12(i=q+1q+rsx¯i2)=(i=q+1q+rσx¯i2Yi2)12(i=q+1q+rσx¯i2Yi),Vq=(i=1qσx¯i2sx¯i4)12=(i=1qσx¯i2Yi2)12,Vr=(i=q+1q+rσx¯i2sx¯i4)12=(i=q+1q+rσx¯i2Yi2)12.
(Uq, Vq) and (Ur, Vr) are mutually independent. As we mentioned before, Uq and Vq represented the initial q subgroups, and hence Ur and Vr represented the remaining r subgroups.

Lemma 2.1.

i=1hσx¯i2Uh2mini=1,2,,h{σx¯i2} holds uniformly in (σx¯12,σx¯22,,σx¯h2) .

Proof.

Due to Cauchy-Schwarz inequality, we have:

Uh2=(i=1hσx¯i2Yi2)1(i=1hσx¯i2Yi)2i=1hσx¯i2.

On the other side,

Uh2=(i=1hσx¯i2Yi2)1(i=1hσx¯i2Yi)2(i=1hσx¯i2Yi2)1(i=1hσx¯i4Yi2)=i=1hσx¯i2Yi2i=1hσx¯i2Yi2σx¯i2mini=1,2,,h{σx¯i2}.

Lemma 2.2.

The inequality E{Uh1}(i=1hσx¯i2)1/2E{Vh} holds uniformly in (σx¯12,σx¯22,,σx¯h2) .

Proof.

Due to the fact:

E{Uh1}=E{(i=1hσx¯i2Yi2)1/2(i=1hσx¯i2Yi)1}(i=1hσx¯i2)1/2
=(i=1hσx¯i2)1/2E{j=1hσx¯j2i=1hσx¯i2Yj1}=(i=1hσx¯i2)1/2E{j=1hσx¯j2i=1hσx¯i2(Yj2)1/2}(i=1hσx¯i2)1/2E{j=1hσx¯j2i=1hσx¯i2Yj2}1/2=E{j=1hσx¯j2Yj2}1/2=E{Vh}.

The inequality (2.1) is based on Lemma 2.1, and inequality (2.2) is due to Jensen’s inequality.

Lemma 2.3.

The inequality (i=1hσx¯i2)E(Uh1)E(Vh1) holds uniformly in (σx¯12,σx¯22,,σx¯h2) .

See [7].

Lemma 2.4.

The probability P{|μˆGD|pμ||μˆGD|qμ|} is 1/π(E{arctan(UrUq1)}+E{arctan(UrUq1+2VrVq1)}) .

Proof.

Without loss of generality, it is clear µ may be assumed to be 0. As p = q + r, we need to compute the probability P{|μˆGD|p||μˆGD|q|} .

By applying the fact that:

μˆGD|p=(i=1qsx¯i2)μˆGD|q+(i=q+1q+rsx¯i2)μˆGD|ri=1q+rsx¯i2,
the probability P{|μˆGD|p||μˆGD|q|} can be written as:
P{|μˆGD|p||μˆGD|q|}=P{(μˆGD|p)2(μˆGD|q)2}=P{((i=1qsx¯i2)μˆGD|q+(i=q+1q+rsx¯i2)μˆGD|ri=1q+rsx¯i2)2(μˆGD|q)2}=P{((i=1qsx¯i2)μˆGD|q+(i=q+1q+rsx¯i2)μˆGD|r)2((i=1q+rsx¯i2)μˆGD|q)2}=2P{μˆGD|q(1+2i=1qsx¯i2i=q+1q+rsx¯i2)μˆGD|rμˆGD|q,μˆGD|q>0}=P{((i=1qσx¯i2sx¯i4)1/2(i=1qsx¯i2)1(i=q+1q+rσx¯i2sx¯i4)1/2(i=q+1q+rsx¯i2)1+2(i=1qσx¯i2sx¯i4)1/2(i=q+1q+rσx¯i2sx¯i4)1/2)((μˆGD|r(i=q+1q+rσx¯i2sx¯i4)1/2(i=q+1q+rsx¯i2)1)/(μˆGD|q(i=1qσx¯i2sx¯i4)1/2(i=1qsx¯i2)1))(i=1qσx¯i2sx¯i4)1/2(i=1qsx¯i2)1(i=q+1q+rσx¯i2sx¯i4)1/2(i=q+1q+rsx¯i2)1}=P{(UrUq1+2VrVq1)μˆGD|r(i=q+1q+rσx¯i2sx¯i4)1/2(i=q+1q+rsx¯i2)1μˆGD|q(i=1qσx¯i2sx¯i4)1/2(i=1qsx¯i2)1UrUq1}.

Note that conditionally given sx¯i2 for i = 1,…, p, the ratio

μˆGD|r(i=q+1q+rσx¯i2sx¯i4)1/2(i=q+1q+rsx¯i2)1μˆGD|q(i=1qσx¯i2sx¯i4)1/2(i=1qsx¯i2)1
follows a Cauchy distribution.

Therefore, the probability in Eq.(2.3), can be denoted as

γ=γ(σx¯12,σx¯22,,σx¯p2)=1/π(E{arctan(UrUq1)}E{arctan(UrUq12VrVq1)})=1/π(E{arctan(UrUq1)}+E{arctan(UrUq1+2VrVq1)}).

3. When is combining preferable ? A Necessary Condition

Theorem 3.1.

P{|μˆGD|pμ||μˆGD|qμ|}1/2 holds uniformly in (μ,σx¯12,σx¯22,,σx¯p2) , only if

2E{(i=q+1pσx¯i2Yi2)1/2(i=q+1pσx¯i2Yi)}E{(i=1qσx¯i2Yi2)1/2(i=1qσx¯i2Yi)1}E{(i=q+1pσx¯i2Yi2)1/2}E{(i=1qσx¯i2Yi2)1/2},
which is equivalent to stating:
2E{Ur}E{Uq1}E{Vr1}E{Vq},
holds uniformly in (σx¯12,σx¯22,,σx¯p2) .

Proof.

According to Lemma 2.4, the probability P{|μˆGD|pμ||μˆGD|qμ|} can be denoted as:

γ=γ(σx¯12,σx¯22,,σx¯p2)=1/π(E{arctan(UrUq1)}+E{arctan(UrUq1+2VrVq1)}).

The expectations are taken with respect to the independent random variables Yi’s.

Let σx¯i2=σx¯12τi , for i = 1,2,…,q; and σx¯i2=σx¯12τi , for i = q+1,q+2,…,q+r. Here τi’s are in the range of [0,∞), except τ1 = 1. a So Uq=σx¯11Uq* , Vq=σx¯1Vq* , Ur=σx¯1Ur* and Vr=σx¯11Vr* , where

Uq*=(i=1qτiYi2)1/2(i=1qτiYi),Vq*=(i=1qτiYi2)1/2,Ur*=(i=q+1q+rτiYi2)1/2(i=q+1q+rτiYi),Vr*=(i=q+1q+rτiYi2)1/2.

The probability in equation (3.2) will be

γ=γ(σx¯12,τ1,τ2,,τp)=1/π(E{arctan(σx¯12Ur*(Uq*)1)}+E{arctan(σx¯12Ur*(Uq*)1+2σx¯12Vr*(Vq*)1)}).

The dominated converge theorem implies that:

γ(σx¯12)1/π(E{arctan0}+E{arctan})=1/2asσx¯120.

Next, we will show that γ ≥ 1/2 uniformly in (σx¯12,σx¯22,,σx¯p2) , only if 2E{Ur}E{Uq1)E{Vq}E{Vr1} holds uniformly in (σx¯12,σx¯22,,σx¯p2) .

The derivative of γ with respect to σx¯12 , is given by

σx¯12πγ=E{Ur*(Uq*)12σx¯14Vr*(Vq*)11+(σx¯12Ur*(Uq*)1+2σx¯12Vr*(Vq*)1)2}+E{Ur*(Uq*)11+σx¯14Ur*2(Uq*)2}=E{Ur*(Uq*)1σx¯142Vr*(Vq*)1σx¯14(1+σx¯14Ur*2(Uq*)2)+4Vr*(Vq*)1(σx¯14Ur*(Uq*)1+Vr*(Vq*)1}+E{Ur*(Uq*)11+σx¯14Ur*2(Uq*)2}.

To check whether the inequality:

E{12Vq*(Vr*)1+Ur*(Uq*)1)}0
is a necessary condition, we assume that inequality (3.4) is not true for some (τ2,…,τp). Then we notice that
σx¯12γ|σx¯12=0=1/πE{12Vq*(Vr*)1+Ur*(Uq*)1)}<0,
which leads to γ being non-decreasing and contradicts that γ ≥ 1/2 for σx¯12 in a small region near 0.

It is easy to verify that inequality (3.4) is equivalent to inequality (3.1).

4. When is combining preferable ? Several Sufficient Conditions

Theorem 4.1.

If

E{21σx¯14Ur*(Uq*)1+Vr*(Vq*)1}+E{Ur*(Uq*)11+σx¯14Ur*2(Uq*)2}0,
then P{|μˆGD|pμ||μˆGD|qμ|}1/2 holds uniformly in (μ,σx¯12,σx¯22,,σx¯p2) .

Proof.

Continue from equation (3.3) in Theorem 3.1, we obtain the following:

σx¯12πγ=E{Ur*(Uq*)1σx¯142Vr*(Vq*)1σx¯14(1+σx¯14Ur*2(Uq*)2)+4Vr*(Vq*)1(σx¯14Ur*(Uq*)1+Vr*(Vq*)1)}+E{Ur*(Uq*)11+σx¯14Ur*2(Uq*)2}E{2Vr*(Vq*)1σx¯14(1+σx¯14Ur*2(Uq*)2)+4Vr*(Vq*)1(σx¯14Ur*(Uq*)1+Vr*(Vq*)1}+E{Ur*(Uq*)11+σx¯14Ur*2(Uq*)2}E{2Vr*(Vq*)14Vr*(Vq*)1(σx¯14Ur*(Uq*)1+Vr*(Vq*)1)}+E{Ur*(Uq*)11+σx¯14Ur*2(Uq*)2}=E{21σx¯14Ur*(Uq*)1+Vr*(Vq*)1}+E{Ur*(Uq*)11+σx¯14Ur*2(Uq*)2}.

Hence if

E{21σx¯14Ur*(Uq*)1+Vr*(Vq*)1}+E{Ur*(Uq*)11+σx¯14Ur*2(Uq*)2}0,
then σx¯12πγ0 , which leads to γ(σx¯12,τ2,,τp) being non-decreasing in σx¯12 for any of (τ2,…,τp). The dominated converge theorem implies that γ(σx¯12)1/π(E{arctan0}+E{arctan})=1/2 as σx¯120 . For any other finite (τ2,…,τp) and σx¯12 , we have γ(σx¯12,τ2,,τp)γ(0,τ2,,τp)=1/2 . At the boundary where one or some τi’s go to ∞, we have γ(σx¯12,τ2,,τp)=1or1/2>1/2 . Hence γ ≥ 1/2 holds uniformly in (σx¯12,σx¯22,,σx¯p2) .

Based on Theorem 4.1, we derive several sufficient conditions in the sequel for P{|μˆGD|pμ||μˆGD|qμ|}1/2 to hold uniformly in (μ,σx¯12,σx¯22,,σx¯p2) .

Corollary 4.2.

2E(Ur)E(Uq1)E(Uq)mini=1,,q(σx¯i2)E(Vr1)
is a sufficient condition for P{|μˆGD|pμ||μˆGD|qμ|}1/2 to hold uniformly in (μ,σx¯12,σx¯22,,σx¯p2) .

Proof.

It is easy to state that

2E(Ur)E(Uq1)E(Uq)mini=1,,q(σx¯i2)E(Vr1)
is equivalent to
2E(Ur*)E(Uq*1)E(Uq*)mini=1,,q(τi)E(Vr*1).

From Theorem 4.1, we only need to show the validity of inequality (4.1).

At the left side of inequality (4.1), we have the following inequality for the second term:

E{Ur*(Uq*)1)1+σx¯14Ur*2(Uq*)2}E{Ur*(Uq*)11+σx¯14(i=q+1q+rτi)(mini=1,,q(τi))1}.

This is because of Ur*2i=q+1q+rτi and (Uq*)2(mini=1,,q(τi))1 following by Lemma 2.1.

We also have:

(1+σx¯14(i=q+1q+rτi)(mini=1,,q(τi))1)E{{σx¯14Ur*(Uq*)1+Vr*(Vq*)1}1}(1+σx¯14(i=q+1q+rτi)mini=1,,q(τi))1E{σx¯14(i=q+1q+rτi)2(mini=1,,q(τi))2(Ur*)1(Uq*)+Vq*(Vr*)1}max{E(Uq*)mini=1,,q(τi)(i=q+1q+rτi)E(Ur*1),E(Vq*)E(Vr*1)}.

The inequality (4.3) follows from Jensen’s inequality:

(1+a)E(aA+B)1(1+a)1E(aA1+B1),
where a=σx¯14(i=q+1q+rτi)mini=1,,q(τi) , A=mini=1,,q(τi)(i=q+1q+rτi)Ur*(Uq*)1 , and B=Vr*(Vq*)1 .

From Lemma 2.3, we know

(i=q+1q+rτi)E(Ur*1)E(Vr*1).

From Lemma 2.2 and the fact that Uq*mini=1,,q(τi)Uq*1 , we note

E(Uq*)mini=1,,q(τi)E(Uq*1)E(Vq*).

So we have the following :

max{E(Uq*)mini=1,,q(τi)(i=q+1q+rτi)E(Ur*1),E(Vq*)E(Vr*1)}E(Uq*)mini=1,,q(τi)E(Vr*1).

Then at the left side of inequality (4.1), we have the following inequality for the first term:

E{{σx¯14Ur*(Uq*)1+Vr*(Vq*)1}1}(1+σx¯14(i=q+1q+rτi)mini=1,,q(τi))1max{E(Uq*)mini=1,,q(τi)(i=q+1q+rτi)E(Ur*1),E(Vq*)E(Vr*1)}(1+σx¯14(i=q+1q+rτi)mini=1,,q(τi))1E(Uq*)mini=1,,q(τi)E(Vr*1)

Combining the above and inequality (4.2), we have

σx¯12πγE{21σx¯14Ur*(Uq*)1+Vr*(Vq*)1}+E{Ur*(Uq*)1)1+σx¯14Ur*2(Uq*)2}{21E(Uq*)mini=1,,q(τi)E(Vr*1)+E{Ur*(Uq*)1}}(1+σx¯14i=q+1q+rτi)mini=1,,q(τi))1.

If

21E(Uq*)mini=1,,q(τi)E(Vr*1)+E{Ur*(Uq*)1}0,
then
σx¯12πγ0.

Therefore a sufficient condition for P{|μˆGD|pμ||μˆGD|qμ|}1/2 to hold uniformly in (μ,σx¯12,σx¯22,,σx¯p2) is:

2E(Ur)E(Uq1)E(Uq)mini=1,,q(σx¯i2)E(Vr1).

Another corollary is given in the following:

Corollary 4.3.

2E(Ur)E(Uq)i=1qσx¯i2E(Vr1),
or
2E(Ur)E(Uq)E(Vq1)E(Vr1).
is a sufficient condition for P{|μˆGD|pμ||μˆGD|qμ|}1/2 to hold uniformly in (μ,σx¯12,σx¯22,,σx¯p2) .

Proof.

We need to validate inequality (4.1) in Theorem 4.1. On the left side of inequality (4.1), we have the following inequality for the second term:

E{Ur*(Uq*)1)1+σx¯14Ur*2(Uq*)2}E{Ur*Uq*i=1qτi+σx¯14(i=q+1q+rτi)}.

It is because of the facts Ur*2i=q+1q+rτi and Uq*2i=1qτi following from lemma 2.1.

We also have:

(i=1qτi+σx¯14(i=q+1q+rτi))E{{σx¯14Ur*(Uq*)1+Vr*(Vq*)1}1}(i=1qτi+σx¯14(i=q+1q+rτi))1E{σx¯14(i=q+1q+rτi)2(Ur*)1(Uq*)+(i=1qτi)2Vq*(Vr*)1}max{(i=q+1q+rτi)E(Ur*)1E(Uq*),(i=1qτi)E(Vq*)E(Vr*)1}.

The inequality (4.5) is from Jensen’s inequality.

So at the left side of inequality (4.1), we have the following inequality for the first term:

E{{σx¯14Ur*(Uq*)1+Vr*(Vq*)1}1}(i=1qτi+σx¯14(i=q+1q+rτi))1max{(i=q+1q+rτi)E(Ur*)1E(Uq*),(i=1qτi)E(Vq*)E(Vr*)1}.

From Lemma 2.3 we have:

(i=q+1q+rτi)E(Ur*1)E(Vr*1).

The upper bound of E(Uq*) and (i=1qτi)E(Vq*) is i=1qτi . This is due to Lemma 2.1 and Lemma 2.2, respectively.

Another upper bound for (i=1qτi)E(Vq*) is E(Vq*1) . Since

(i=1qτi)E(Vq*)=(i=1qτi)E(Uq*1)E(Vq*)E(Uq*1)E(Vq*1)E(Vq*)E(Uq*1)E(Vq*1),
then we have
max{(i=q+1q+rτi)E(Ur*)1E(Uq*),(i=1qτi)E(Vq*)E(Vr*)1}i=1qτiE(Vr*1),
or
max{(i=q+1q+rτi)E(Ur*)1E(Uq*),(i=1qτi)E(Vq*)E(Vr*)1}max{E(Vq*1),E(Uq*)}E(Vr*1)=E(Vq*1)E(Vr*1).

Combining inequalities (4.4), (4.6) and (4.7), we have the following:

σx¯12πγE{21σx¯14Ur*(Uq*)1+Vr*(Vq*)1}+E{Ur*(Uq*)1)1+σx¯14Ur*2(Uq*)2}{21i=1qτiE(Vr*1)+E(Ur*)E(Uq*)}(i=1qτi+σx¯14(i=q+1q+rτi))1,
while combining inequalities (4.4), (4.6) and (4.8), we obtain
σx¯12πγE{21σx¯14Ur*(Uq*)1+Vr*(Vq*)1}+E{Ur*(Uq*)1)1+σx¯14Ur*2(Uq*)2}{21E(Vq*1)E(Vr*1)+E(Ur*)E(Uq*)}(i=1qτi+σx¯14(i=q+1q+rτi))1.

So if

21i=1qτiE(Vr*1)+E(Ur*)E(Uq*)0,
or
21E(Vq*1)E(Vr*1)+E(Ur*)E(Uq*)0,
then
σx¯12πγ0

Hence it is equivalent to saying that a sufficient condition for P{|μˆGD|pμ||μˆGD|qμ|}1/2 to hold uniformly in (μ,σx¯12,σx¯22,,σx¯p2) is:

2E(Ur)E(Uq)i=1qσx¯i2E(Vr1),
or
2E(Ur)E(Uq)E(Vq1)E(Vr1).

5. Sample Size Discussion

In this section we will discuss the sample size requirement to ensure the sufficient condition provided in Corollary 4.3:

2E(Ur)E(Uq)E(Vq1)E(Vr1),
Which is the same as:
2E{(i=1qθiYi)(i=1qθiYi2)12}E{(i=q+1q+rθiYi)(i=q+1q+rθiYi2)12}E(i=1qθiYi2)12E(i=q+1q+rθiYi2)12,
θi=σx¯i2 , i = 1,2,…, p. We specified the following theorem:

Theorem 5.1.

Let Y1,…,Yp be independent random variables such that Yi~miχ2(mi) , for i = 1,2,…, p,and let p = q + r. Then for any θi > 0 (i = 1,2,…,p),

2E{(i=1qθiYi)(i=1qθiYi2)12}E{(i=q+1q+rθiYi)(i=q+1q+rθiYi2)12}E(i=1qθiYi2)12E(i=q+1q+rθiYi2)12
if there exists 1 < b < 2, such that
mi1(mi4)+2(j=1qmj)11b
for all i = 1,2,…,q, and
mi1(mi4)+2(j=q+1q+rmj)1b2
for all i = q + 1,q + 2,…,q + r = p.

Proof.

Define:

g1(Yi)=E{Yi(j=1qθjYj2)12|Yi},
for i = 1,2,…,q; and define:
g2(Yi)=E{Yi(j=q+1q+rθjYj2)12|Yi},
for i = q + 1,q + 2,…,q + r = p.

From [7] and [2], we know the following for i = 1,2,…,q:

miE(g1(Yi))=(mi2)E(Yig1(Yi))2E(Yi2g1(Yi)),
and similarly for i = q + 1,q + 2,…,q + r = p, we have:
miE(g2(Yi))=(mi2)E(Yig2(Yi))2E(Yi2g2(Yi)).
Then,
g1(Yi)=E{(j=1qθjYj2)12|Yi}θiE{Yi2(j=1qθjYj2)32|Yi},
and
g2(Yi)=E{(j=q+1q+rθjYj2)12|Yi}θiE{Yi2(j=q+1q+rθjYj2)32|Yi}.

For g1(Yi), we have that:

miE{Yi(j=1qθjYj2)12}=miE(g1(Yi))=(mi2)E(Yig1(Yi))2E(Yi2g1(Yi))=(mi4)E{Yi2(j=1qθjYj2)12}+2θiE{Yi4(j=1qθjYj2)32}.

Based on the above, we see the following

E{(i=1qθiYi)(i=1qθiYi2)12}=i=1qθiE{Yi(j=1qθjYj2)12}=E{(i=1qmi1(mi4)θiYi2)(i=1qθiYi2)12}+2E{(i=1qmi1θi2Yi4)(i=1qθiYi2)32}.

From Cauchy-Schwarz inequality, we can tell:

i=1qmi1θi2Yi4(i=1qmi)1(i=1qθiYi2)2.

Applying (5.3) into (5.2), we have the following:

E{(i=1qθiYi)(i=1qθiYi2)12}E{[i=1q(mi1(mi4)+2(j=1qmj)1)θiYi2](i=1qθiYi2)12}.

Similarly for g2(Yi), we have that:

E{(i=q+1q+rθiYi)(q+i=1q+rθiYi2)12}E{[i=q+1q+r(mi1(mi4)+2(j=q+1q+rmj)1)θiYi2](i=q+1q+rθiYi2)12}.

Thus, inequality (5.1) is true if there exists 1 < b < 2, such that

mi1(mi4)+2(j=1qmj)11b
for all i = 1,2,…,q, and
mi1(mi4)+2(j=q+1q+rmj)1b2
for all i = q + 1,q + 2,…,q + r = p.

Remark 1

Observe that inequality (5.4) and inequality (5.5) are respectively equivalent to:

(11b){mmin(mmin211b)+(j=1,jminqmj)(mmin411b)}0
and
(1b2){mmin(mmin21b2)+(j=q+1,jminq+rmj)(mmin41b2)}0.

In order to have solutions for inequalities (5.6) and (5.7), we need 1 < b < 2.

Remark 2

One symmetric solution is to let b=2 . Check the Condition (5.6),

1. (1)

When q = 1, Condition (5.6) holds if m14227 .

2. (2)

When q = 2, Condition (5.6) holds if one of the following holds:

1. (a)

mmin = 11, and the other 11 ≤ mj,jmin ≤ 17

2. (b)

mmin = 12, and the other 12 ≤ mj,jmin ≤ 37

3. (c)

mmin = 13, and the other 13 ≤ mj,jmin ≤ 122

4. (d)

mi ≥ 14 for i = 1,2.

3. (3)

When q = 3, Condition (5.6) holds if one of the following holds:

1. (a)

mmin = 12, and j=13mj,jmin37 . So either all are equal to 12 or some mi = 13 and the other two are each equal to 12.

2. (b)

mmin = 13, and j=13mj,jmin122

3. (c)

mi ≥ 14 for all i = 1,2,3.

4. (4)

When q = 4, Condition (5.6) holds if one of the following holds:

1. (a)

mmin = 13, and j=14mj,jmin122

2. (b)

mi ≥ 14 for all i = 1,2,3,4.

5. (5)

When q = 5, Condition (5.6) holds if one of the following holds:

1. (a)

mmin = 13, and j=15mj,jmin122

2. (b)

mi ≥ 14 for all i = 1,2,3,4,5.

6. (6)

When q = 6, Condition (5.6) holds if one of the following holds:

1. (a)

mmin = 13, and j=16mj,jmin122

2. (b)

mi ≥ 14 for all i = 1,2,3,4,5,6.

7. (7)

When q = 7, Condition (5.6) holds if one of the following holds:

1. (a)

mmin = 13, and j=17mj,jmin122

2. (b)

mi ≥ 14 for all i = 1,2,3,4,5,6,7.

8. (8)

When q = 8, Condition (5.6) holds if one of the following holds:

1. (a)

mmin = 13, and j=18mj,jmin122

2. (b)

mi ≥ 14 for all i = 1,2,3,4,5,6,7,8.

9. (9)

When q = 9, Condition (5.6) holds if one of the following holds:

1. (a)

mmin = 13, and j=19mj,jmin122

2. (b)

mi ≥ 14 for all i = 1,2,3,4,5,6,7,8,9.

10. (10)

When q ≥ 10, Condition (5.6) holds if mi ≥ 14 for all i = 1,2,…,q.

Similar discussion can be found for condition (5.7).

6. Conclusion

In this paper we compared the p-source based GDE with its q-sub-source based competitors under Pitman closeness criterion. We established a necessary condition and several sufficient conditions for the p-source based GDE to be Pitman closer than its q-sub-source based GDE. We further discussed the sample size requirement corresponding to each source, and we found that one sufficient condition is ni ≥ 15 for i = 1,2,…, p.

This sample size requirement is relatively close to the requirement based on mean square errors. Hence in our point of view, the p-source-based GDE dominates any other q-sub-source based GDE not only in terms of mean square error loss function but also in the sense of Pitman closeness criterion of probability coverage, when the sample size of each source is moderately large enough.

Acknowledgements

Research is supported by the U.S. National Science Foundation Grants DMS-1306394 and DMS-1546674. This work was undertaken during regular visits of Bikas.K. Sinha at University of Illinois, Chicago. He is a retired professor of Indian Statistical Institute, Kolkata. The authors are extremely thankful to two anonymous referees for critically reading the manuscript and offering numerous suggestions towards clearer presentation of our ideas.

Footnotes

a

In inequality (3.4), we are going to prove, 2E{Ur}E{Uq1}E{Vq}E{Vr1} , it will follow that U’s and V1s are proportional to σ2 .

Journal
Journal of Statistical Theory and Applications
Volume-Issue
17 - 2
Pages
291 - 306
Publication Date
2018/06/30
ISSN (Online)
2214-1766
ISSN (Print)
1538-7887
DOI
https://doi.org/10.2991/jsta.2018.17.2.9How to use a DOI?
Open Access
This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

TY  - JOUR
AU  - Keyu Nie
AU  - Bikas.K. Sinha
AU  - A.S. Hedayat
PY  - 2018
DA  - 2018/06/30
TI  - Performance of the Graybill–Deal Estimator via Pitman Closeness Criterion
JO  - Journal of Statistical Theory and Applications
SP  - 291
EP  - 306
VL  - 17
IS  - 2
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.2018.17.2.9
DO  - https://doi.org/10.2991/jsta.2018.17.2.9
ID  - Nie2018
ER  -