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].
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¯i2∑i=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 j ≠ i). [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{|μˆ1−μ|≤|μˆ2−μ|}≥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|2−μ|≤|x¯i−μ|}≥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|p−μ|≤|x¯i−μ|}≥1/2,
for all i = 1,2,…, p and uniformly in
(μ,σx¯12,σx¯22,…,σx¯p2)
, if and only if
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
1−8mk−1+4(∑j=1,j≠ipmj)−1≥0,
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.
(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.
The expectations are taken with respect to the independent random variables Yi’s.
Let
σx¯i−2=σx¯1−2τi
, for i = 1,2,…,q; and
σx¯i−2=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¯1−1Uq*
,
Vq=σx¯1Vq*
,
Ur=σx¯1Ur*
and
Vr=σx¯1−1Vr*
, where
then
∂∂σx¯12πγ≥0
, 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¯12→0
. 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|p−μ|≤|μˆGD|q−μ|}≥1/2
to hold uniformly in
(μ,σx¯12,σx¯22,…,σx¯p2)
.
Corollary 4.2.
2E(Ur)E(Uq−1)≥E(Uq)mini=1,…,q(σx¯i−2)E(Vr−1)
is a sufficient condition forP{|μˆGD|p−μ|≤|μˆGD|q−μ|}≥1/2to hold uniformly in(μ,σx¯12,σx¯22,…,σx¯p2)
.
Proof.
It is easy to state that
2E(Ur)E(Uq−1)≥E(Uq)mini=1,…,q(σx¯i−2)E(Vr−1)
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:
When q = 2, Condition (5.6) holds if one of the following holds:
(a)
mmin = 11, and the other 11 ≤ mj,j≠min ≤ 17
(b)
mmin = 12, and the other 12 ≤ mj,j≠min ≤ 37
(c)
mmin = 13, and the other 13 ≤ mj,j≠min ≤ 122
(d)
mi ≥ 14 for i = 1,2.
(3)
When q = 3, Condition (5.6) holds if one of the following holds:
(a)
mmin = 12, and
∑j=13mj,j≠min≤37
. So either all are equal to 12 or some mi = 13 and the other two are each equal to 12.
(b)
mmin = 13, and
∑j=13mj,j≠min≤122
(c)
mi ≥ 14 for all i = 1,2,3.
(4)
When q = 4, Condition (5.6) holds if one of the following holds:
(a)
mmin = 13, and
∑j=14mj,j≠min≤122
(b)
mi ≥ 14 for all i = 1,2,3,4.
(5)
When q = 5, Condition (5.6) holds if one of the following holds:
(a)
mmin = 13, and
∑j=15mj,j≠min≤122
(b)
mi ≥ 14 for all i = 1,2,3,4,5.
(6)
When q = 6, Condition (5.6) holds if one of the following holds:
(a)
mmin = 13, and
∑j=16mj,j≠min≤122
(b)
mi ≥ 14 for all i = 1,2,3,4,5,6.
(7)
When q = 7, Condition (5.6) holds if one of the following holds:
(a)
mmin = 13, and
∑j=17mj,j≠min≤122
(b)
mi ≥ 14 for all i = 1,2,3,4,5,6,7.
(8)
When q = 8, Condition (5.6) holds if one of the following holds:
(a)
mmin = 13, and
∑j=18mj,j≠min≤122
(b)
mi ≥ 14 for all i = 1,2,3,4,5,6,7,8.
(9)
When q = 9, Condition (5.6) holds if one of the following holds:
(a)
mmin = 13, and
∑j=19mj,j≠min≤122
(b)
mi ≥ 14 for all i = 1,2,3,4,5,6,7,8,9.
(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.
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 - 10.2991/jsta.2018.17.2.9
ID - Nie2018
ER -
