Paul Chiou. Lamar University, Beaumont. C.P.Ban. The University of Texas, Arlington ..... University and Louisiana Tech University. He received his MA and PhD ...
IEEE TRANSACTIONS ON REUABll.ITY, VOL. 38, NO.4, 1989 OCTOBER
449
Shrinkage Estimation of Threshold Parameter of the Exponential Distribution Paul Chiou
~Pt
Lamar University, Beaumont
~s
preliminary test estimator of 1/ upper a Cdf point of F distribution with 2 and 2 (r - I ) degrees of freedom preliminary test shrinkage estimator of 1/
k
shrinkage coefficient
C
C.P.Ban The University of Texas, Arlington
a
Key Words - Preliminary test estimator, Shrinkage estimator, Bias, Mean square error, Minimax regret criterion, Effective interval
s-significance level B{· } s-bias MSE { . } mean square error Ri { • ; .} risk function REG { . ; . } regret function
Reader Aids Purpose: Widen state of the art Special math needed for explanations: Statistical theory Special math needed to use results: Same Results nseful to: Reliability theoreticians
3. SHRINKAGE ESTIMATORS OF THRESHOLD PARAMETER 3.1 A preliminary
Abstract - This paper studies the usual preliminary test estimator of the threshold parameter of the exponential distribution in censored samples. The optimal levels of significance arid their corresponding critical values for the preliminary test are obtained. The optimal values of shrinkage coefficients for a preliminary test shrinkage estimator are also obtained based on the minimax regret criterion.
I. INTRODUCTION The exponential distribution has been used widely in the field of life testing and reliability theory. Epstein & Sobel [4] obtained the minimum variance unbiased estimator (MVUE) for its scale parameter and threshold parameter respectively. The shrinkage estimators of the scale parameter have been proposed by Bhattacharya & Srivastava [2], and Pandey [8]. In this paper a preliminary test estimator of the threshold parameter is studied. The optimal levels of s-significance and their corresponding critical values for the preliminary test are obtained by using the minimax regret criterion [3, 7, 9, to]. We also consider a preliminary test shrinkage estimator [see (3.17)]. Following the procedure in Inada [6], we obtain the optimal values of shrinkage coefficients for the preliminary test shrinkage estimator. These two estimators are compared in section 4. Section 5 is a numerical example.
1/
8
Xi 1; 1/0
Let Xi' i = 1,... .r denote the first r ordered observations in a sample of size n from the 2-parameter exponential distribution with survival function: = exp[ - (x-'I1)/8],
S(X)
threshold parameter of exponential distribution scale parameter of exponential distribution first r order statistics in sample of size' n, i 1,2, ... r minimum variance unbiased estimator of 1/ prior point estimate of 1/
X ~ 1/,
'11
0, 8
~
>
O.
The T == Ei=l (Xi -Xl) + (n-r) (Xr - Xl) and Xl are s-independent [4]. Moreover, ~ = Xl - T/[n(r-l)] is the MUVE of 1/. Since 2n(Xl - 1/)/8 and 2T/() have chi-square distributions with 2 and 2 (r - 1) degrees of freedom respectively, the statistic :f = n(r-l )(Xl -1/0) /Tis used for testing the preliminary hypothesis Hp : 1/ =1/0' The preliminary test estimator for 1/ is:
• 1/pt
=
[ 1/00 if
0
:5 n(r-1)(XI-1/0)/T
C
~
(3.1)
~, otherwise. The mean of ijpt is:
E{~Pt}
=
E{1/o
+
(1]0-1])}
+
(1/0-1/)}
(XI-1/)
2. NOTATION
test estimator
>
I
(1/0-'11)
Pr{
+
(1]0-1])
E{~
TC/[n(r-l)]
I
(XI-1/)
:5 :5
(XI-1])
(XI-1/)
+
:5 TC/[n(r-l)]
-
{[r/ (r-l
[k(k-2)a/n2]
1
i.
=
+
2[Cdr
+
[Cdr +dr-I
(2kl/;a/n) )][C2dr+1
dr-I
-
(3.18)
'I/o
= r/[n'(r-l)]
-2(k~bln)
[ (i-d)
E
