On shrunken estimators for exponential scale parameter

Journal

of Statistical

Planning

and Inference

24 (1990) 87-94

87

North-Holland

ON SHRUNKEN ESTIMATORS SCALE PARAMETER N.S. KAMBO, Department

B.R. HANDA

of Mathematics,

Indian

FOR EXPONENTIAL

and Z.A. Institute

Al-HEMYARI

of Technology,

Hauz

Khas,

New Delhi

110016,

India Received

19 August

Recommended

Abstract:

1987; revised manuscript

by M.L.

This article considers

value Be for 0 is available. sample

received

29 March

1988

Puri

a shrunken

The estimator

of size n. The estimator

is shown

estimator is based

of the exponential on the first r ordered

to be more efficient

mean life 0 when a guess observations

out of a

than the usual estimators

when tJ

is close to 00. AMY Key

Subject words

squared

Classification: and phrases:

62F10.

Shrunken

estimator;

exponential

mean

life; censored

sample;

mean

error.

1. Introduction Let X(,)rX(,)r ... IX(,) size n from an exponential

be the first r ordered density

f(x, 19)= 8-l exp(-x/B),

x >O,

statistics

of a random

19> 0.

sample

of

(1)

Based on this censored sample, let e be a ‘good estimator’ for 8. Suppose a prior guess value B0 for B is available. According to Thompson (1968) 0, is a ‘natural origin’ and such natural origins may arise for any one of a number of reasons, e.g., we are estimating 19and: (i) we believe 8, is close to the true value of 0; or (ii) we fear that 0, may be near the true value of 8, i.e., something bad happens if 0 = 8,, , and we do not know about it. For such cases, Thompson suggested a shrinkage technique of moving 6 close to &, thus obtaining an estimator which is better than e near Q,, though possibly worse farther away, measured in terms of mean squared error. Several authors studied shrinkage estimators for the mean 0 of exponential density (see, e.g., Pandey and Singh (1980), Pandey (1983), Hirano (1984) and the references contained therein). In this paper we study an estimator for f3 given by 0378.3758/90/$3.50

% 1990, Elsevier

Science Publishers

B.V. (North-Holland)

88

N.S. Kambo et al. / Shrunken estimators for scale parameter

&T=

k(B- t9,) + e,

if PER,

(I-k)(&B,)+&

if d$R,

(2)

where 0 i kl 1 and R is a suitably chosen region in the parameter space { 8: 0> 0). Methods of choosing R and k are discussed. The expressions for the bias and the mean squared error (MSE) of e” are given when 6 is MLE (maximum likelihood estimator). Some numerical results are presented to show superiority of &over usual estimators when B is moderately close to 0,.

2. Choice

of region R

Three choices for the region R, which are independent ly, let the region R, be defined by

of k, are considered.

First-

R, = {e : (e- eo)2 I MSE(&‘)}.

(3)

Sometimes it may be possible to express R, as an interval. is not possible, we may approximate R, by the interval

If such a simplification

R, = { 6 : (0 - ~9,)~I MSE(&B,)} = [max(O, e. - 1/MsE(8/eo)), e. + flGiSE(e/e,)l. This gives the second choice. significance (x given by R, = @:

The third

choice

is the pretest

(4) region

W%V-.,,A-,211,

of level of

(9

where I, _a,2 and ~4,~~are the lower and upper lOOa/2-percentile points of the test statistic T used for testing H,: 6’= B0 against the alternative H, : B # 0,. The region and Srivastava (1974), R3 was used by many authors see, e.g., Bhattacharya Pandey and Singh (1980).

3. Bias and mean squared error of estimator

#

The bias and MSE of the shrunken estimator &for any region R can be expressed in terms of the bias B(&/B) and MSE(8/8) of 8 as B(B/e, R) = (I-

k)iqB/e) + k(e, - e) - (I-

2k)pul(O/e,R),

(6)

and MSE(&0,

R) = (1 - k)2 MSE(&B) + k2(0 - 0,)2 - 2k(i - k)(e - e,)~(fVe)

- (I-

+ 2(1- 2k)(e - e,)p@e,

R),

2k)p2@e, R) (7)

N.S. Kambo et al. / Shrunken estimators for scale parameter

where

^ Pi(S/S,R)

=

E[(@-

=

I,(@ and j(e/e)

=

(6

a eCIY1~(e)l

13&o/0)

1

if $ER,

0

otherwise,

is the pdf of the estimator

Remark 1. When

de,

i = 1,2,

(8)

6.

0 = f&,, we have from

MSE(8/8,,

89

(7), = -k(l

R) - MSE(&&)

- k)MSE(&B,,)

- k[MSE@B,)

- (1 - k),u2(&S0, R)

-j.&/BO,

R)] I 0,

for any region R, estimator 0, and k, 05 ks 1. This shows that shrinkage behaves better when 0 is sufficiently close to 0,.

4. Choice

estimator

of k

Suppose we select k that minimizes zero, we get

MSE(8/0,

R). Setting

(a/ak)MSE(8/8,

R) to

” k = k2 =

MSE(8/8) i[

-

(s!R

0)2f(&9)

+(e-eo)* ’ _f(e/e) dQ+(eI .R

/{hm(B/e)

1

df?

e,)iqe/e)

1

+ (e - eo)2 + 2(e - e,)B(B/e) 1.

It is easily seen that the denominator of RHS of (9) is equal to E(& (J2/ak2)MSE(&/t9, R) = 2E(0- &J2 2 0, it follows that the minimizing kE [0, l] is given by 0 k” =

k2

1

I

ifk,
(9) B0)2. Since value of

1,

In case 6 is an unbiased estimator of 8 (e.g. MLE of the next section), it follows from (9) that k2 E [0, l] and hence k* = k, . Also, by putting 8 = 0, in (9), we see that k

=

1

1

_

iu2(Q4,R)E]O,l] MSE(&‘8,)

(10)

minimizes MSE(@/&, R). One disadvantage of choice k* over k, is that the former is a function of the unknown parameter 8.

90

N.S. Kambo et al. / Shrunken estimaiors for scale parameter

5. Shrinkage

estimator

based on MLE

Let &t be the shrunken

estimator

obtained

from (2) when 6 is taken

as the MLE

8= T,/r,

(11)

where T, = i

XCi, f (n -

r)A$,.

(12)

i=l

It is known that E(B)= 8, Var(t!J/f3)=MSE(B/6’) = e2/r and 2&/B follows a chisquare distribution with 2r degrees of freedom. Suppose statistic T= 2rB/e, is used to test H,, : O= Bo. Denote by R,(l) the region corresponding to R; (i= 1,2,3) of Section 2. It can be easily seen from (3), (4) and (5) that

and

where xf_a,2,2r and &2r are the lower and upper 100 cr/2-percentile chi-square distribution with 2r degrees of freedom. Writing (1) Ri = ]&a,+ ~&I, simple

calculations

points

of the

i = 1,&3,

(14)

show that

,ul(8/e,R!‘)) = e[G(2r+2; Biybj)-AG(2r; a,,6J],

(15)

fiuz(fve,R!l))= e2 -r+ ’ G(2r+ 4; Diy6i)-2AG(2r+ 2; pi, pi) [ r + A2G(2r;cti,bi) ,

(16)

1

where pi -=ai(n)

= 2rAQi,

5i s hi(A) =

2rAbi,

i = e,/e, (17)

G(m; x,_Y) = ‘%W-G,(x), and G,(x) is the distribution function of a chi-square random variable with m degrees of freedom. Using these results in (6) and (7), the expressions for bias B(C&/0, RF’))and mean squared error MSE(B, /f3, RI”) of gl are, respectively, given by and MSE(B,/B,

(1-k)’

Rj’)) = e* ___ r

+k2(1-A)2-(1-2k)

-2G(2r+2;

r+l G(2r+4; ~ii,6;) i r __

di, 6i) + A(2_A)G(2r;

.

0i, bi) II

(19)

91

N.S. Kambo ei al. / Shrunken estimators for scale parameter

6. Numerical Define

results and conclusions

the relative EFF(B,;

efficiency

of an estimator

0, with respect

to estimator

& by

82) = MSE(B,)/MSE(&.

Since the shrunken estimator ti, is biased, it is better to compute EFF(&,; or), where 8r = T,/(r+ 1) is the minimum mean squared error estimator of 8 having MSE equal to 02/(r+ 1). The bias ratio, B(8,/tY,Rj’))/B, obtained from relation (18), and the efficiency EFF(@,; 8,) were computed for r=4(2)12, A =BO/B=O.OO1, 0.005, 0.01, 0.05,0.1(0.1)1.6, 2.0, 2.5, 3.0,4.0 and a=O.Ol, 0.05, 0.15 for the three types of regions I?!‘), I‘-- 1,2,3, and using two values of k, namely, k, and k2. Some of these results are presented in Tables 1 and 2, where the bias ratio is shown in braces. These results compare quite favourably with those given in Bhattacharya and Srivastava (1974). Based on an information criterion, Hirano (1984) suggested taking a = 0.15 when the pretest region I?$‘) is used. Our computations showed that when 0~8, and the pretest region I?:‘) is used, EFF(g,; 0,) does not change much with a. However, when 19=&, EFF(gr; 8,) at a-0.15 is much higher than at cx=O.Ol or 0.05. It was also observed from computations that EFF(8r; 0,) was of the same order for regions I?!‘) and Rir’ For k = k2, the region Ri’) (a = 0.15) yields better efficiency than region R,(‘) if b .9sAs3.0 and r>4. The value k=k, compares quite favourably with k = k2 for 0.8 IA 5 1.2. We may note that EFF(f?,; 0) =((r+ 1)/r)) EFF(g,; 0,). Thus values of relative efficiency of Qr with respect to MLE 8 can be obtained from Tables 1 and 2 by multiplying the efficiency entries by (r+ 1)/r, which is greater than unity.

Table

1. EFF(&, &) and B(@,

18,Ri”) 10 for k = k2 and region R\‘) r

i 0.001

4

0.01

0.05

0.1

1.ooo

10

1.000

12 1.ooo

(-0.111)

(-0.091)

( - 0.200)

1 .OOl (-0.143)

1.OOl (-0.112)

(-0.091)

1.004 (-0.201)

1.002

1.002

( - 0.148)

(-0.112)

(~ 0.092)

(- 0.078)

1.044 (-0.206)

1.015 (-0.148)

1.012 (-0.116)

(- 0.095)

1.033 (-0.154)

1.026 (-0.120)

1.021

1.018

(~ 0.099)

(- 0.084)

1.071

1.060 (-0.131)

1.050 (-0.108)

( - 0.092)

1.002

1.082 (~0.213)

0.2

8

1.000 (-0.143)

1.000

(- 0.200) 0.005

6

1.390

(~ 0.238)

1.003

(-0.194)

1.002

1.001

1.010

(~ 0.077) 1.001

(~ 0.077)

1.008

(- 0.080)

1.043

92 Table

N.S. Kambo et al. / Shrunken estimators for scale parameter 1 (continued) r

A 0.3

0.4

4 1.312

1.099

(- 0.279)

(- 0.229)

1.312 (-0.303)

0.5

1.710 (-0.288)

0.6

2.414 (-0.233)

0.7

3.278 (-0.159)

0.8

3.278 (-0.085)

0.9

3.808 (-0.023)

1.0

1.1

1.2

1.3

1.4

1.5

1.6

2.0

3.0

6

1.164

(- 0.249)

8 1.089 (-0.149) 1.188 (-0.179)

1.371

1.230

(- 0.249)

(- 0.209)

1.868 (-0.231) 2.810 (-0.177) 3.964 (-0.105)

1.572 (-0.213) 2.359 (-0.178) 3.713 (-0.114)

4.504

4.812

(0.035)

(0.043)

10 1.081 (-0.121) 1.102 (-0.144) 1.162 (-0.176) 1.401 (-0.193) 2.046 (- 0.176) 3.393 (-0.119) 4.915 ( - 0.049)

12 1.074 (-0.103) 1.096 (-0.121) 1.128 (-0.150) 1.296 (-0.174) 1.825 (-0.167) 3.096 (-0.120) 4.908 (-0.052)

3.545

4.178

4.583

4.861

5.061

(0.028)

(0.024)

(0.020)

(0.017)

(0.014)

3.188

3.516

3.670

3.733

3.744

(0.071)

(0.074)

(0.072)

(0.071)

(0.069)

2.778

2.827

2.766

2.664

2.549

(0.111)

(0.116)

(0.116)

(0.114)

(0.113)

2.353

2.228

2.068

1.914

1.779

(0.150)

(0.154)

(0.152)

(0.148)

(0.144)

1.963

1.761

1.583

1.441

1.329

(0.186)

(0.186)

(0.179)

(0.171)

(0.163)

1.636

1.422

1.265

1.153

1.074

(0.217)

(0.210)

(0.197)

(0.183)

(0.169)

1.377

1.184

1.060

0.982

0.933

(0.243)

(0.226)

(0.205)

(0.183)

(0.164)

0.797 (0.131)

0.830 (0.104)

0.839

0.778

0.775

(0.286)

(0.221)

(0.169)

0.664

0.780

0.870

0.916

0.937

(0.182)

(0.099)

(0.065)

(0.050)

(0.041)

Remark 2. Let & be the estimator e when 6 in (2) is taken as the minimum mean squared error estimator O1. The expressions for the bias and MSE of & can be easily obtained. We computed EFF(&; e,) for various values of r, A and (x considered above and using k, and k, and the three types of regions. It was observed that

N.S. Kambo et al. / Shrunken estimators for scale parameter Table

2. EFF(&,;

&) and B(~,/R,R~‘))/B

for k=kz

and region

93

Ri” with a=0.15

r A 0.3

4 1.350 (-0.274)

0.4

0.5

0.6

1.365

1.600 ( - 0.275)

(-0.213)

2.010

2.527

2.965 ( - 0.104)

0.9

3.179 (-0.045)

1.0

1.1

1.2

1.3

1.4

1.5

1.6

2.0

3.0

4.0

1.219 (-0.310)

(-0.164) 0.8

0.873 (- 0.338)

(-0.275)

(-0.227) 0.7

6

1.707

1.888 (-0.165) 2.239 (-0.159) 3.582 (-0.143) 9.887 (-0.088)

8

0.991 (-0.288) 1.345 (-0.159) 1.374 (-0.160) 1.587 (-0.175) 2.122 (-0.174) 3.665 (-0.151) 12.000

10 1.170 (-0.131) 1.171 (-0.130) 1.273 (-0.143) 1.477 (-0.154) 1.919 (-0.158) 3.182 (-0.143) 10.000

12 1.088 (-0.102) 1.137 (-0.113) 1.231 (-0.125) 1.404 (-0.137) 1.778 (-0.144) 2.846 (-0.135) 8.615

(-0.092)

(-0.091)

(~ 0.089) 1137211~10’~

3.208

65.111

786391.100

4101141x106

(0.020)

(0.001)

(0.000)

(0.000)

3.085

14.118

12.000

10.000

8.615

(0.090)

(0.094)

(0.093)

(0.091)

(0.089)

(0.000)

2.717

4.407

3.661

3.182

2.846

(0.154)

(0.161)

(0.152)

(0.143)

(0.135)

2.356

2.444

2.123

1.919

1.778

(0.201)

(0.195)

(0.174)

(0.158)

(0.144)

1.958

1.750

1.583

1.417

1.404

(0.229)

(0.204)

(0.175)

(0.154)

(0.137)

1.647

1.429

1.333

1.273

1.231

(0.239)

(0.200)

(0.167)

(0.143)

(0.125)

1.425

1.254

1.197

1.162

1.137

(0.199)

(0.190)

(0.154)

(0.130)

(0.113)

1.023

1.000

1.000

1.000

1.000

(0.150)

(0.143)

(0.100)

(0.091)

(0.077)

0.850

0.893

0.917

0.932

0.942

(0.117)

(0.080)

(0.061)

(0.048)

(0.041)

0.822 (0.081)

0.873

0.901 (0.041)

0.919

0.932

(0.055)

(0.033)

(0.028)

EFF(&; 0,) is generally more than EFF(6,; 0,) for A10.9. Also, when k= k2 and the pretest region R3 (a= 0.15) are used, it was seen that EFF(&; 8,) is greater than

94

N.S. Kambo et al. / Shrunken estimators for scale parameter

unity for 0.552 14.0 and moreover values of EFF(t&; given in Tables 2 and 3 of Pandey and Singh (1980).

0,) are higher

than

those

Acknowledgements We are thankful to the referee for his valuable improvement of the paper.

suggestions

that led to considerable

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(1974). A preliminary

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when the selection

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estimation

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IEEE Trans. Reliability

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B.N. and P. Singh (1980). Shrinkage samples.

Thompson,

J.R.

63, 113-123.

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