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In this paper, we consider estimation of mean time to failure given the exponential model using "two parts" loss functions, namely balanced loss function (BLF) ...
Statistical Papers

Statistical Papers49, 121-131 (2008)

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Estimation of the exponential mean time to failure under a weighted balanced loss function A. A s g h a r z a d e h 1, N. Sanjari F a r s i p o u r 2 i Department of Statistics, Faculty of Basic Science, University of Mazandaran, Post code 47416-1467, Babolsar, Iran; (e-mail: [email protected]) 2 Department of Statistics, College of Sciences, Shiraz University, Shiraz 71454, Iran Received: April 28, 2004; revised version: October 4, 2005 Abstract

This paper considers estimation of a exponential mean time to failure using a loss function that reflects both goodness of fit and precision of estimation. The admissibility and inadmissibility of a class of linear estimators of the form cX + d are studied. K e y w o r d s : Admissibility; Balanced loss function; Bayes estimtor; Blyth's method; Inadmissibility;.

1

Introduction

The exponential distribution plays an i m p o r t a n t role in life testing experiments. In life testing research the most widely used model is the exponential distribution, Exp(0)-distribution, with the p.d.f, of the failure time X is given by 1 f(xlO ) = ~ e - ~ ,

x>0,

0>0.

(1.1)

where 0 is the mean time to failure and it also acts as a scale parameter.

122

In this paper, we consider estimation

of mean

time to failure

given the exponential model using "two parts" loss functions, namely balanced

loss function (BLF)

and weighted

(WBLF),

that reflect both goodness

balanced

loss function

of fit and precision of estima-

tion. The loss functions used in the most of the literature on decision -theoretic analysis are the quadratic loss function and its variants. However,

as discussed in Berger (1985, pp. 60-64), there are several

non-quadratic

loss functions, and the non-quadratic

have also been used in many

loss functions

studies. For example, James and Stein

(1961), Haft (1979, 1980) and Yang (1992) used the entropy loss functions, and Varian (1975), Zellner (1986), Parsian (1990) and Parsian and Sanjari Farsipour (1993) used asymmetric

LINEX

loss functions.

The loss functions used so far, including the LINEX

loss func-

tions, place sole emphasis on the precision of an estimator and ignore issues such as goodness

of fit. However,

one may

have interest in

goodness of fit of the overall model as well as precision of estimation. From

this viewpoint,

Zellner (1994) proposed

the BLF

which takes

account of both "goodness of fit" and "precision of estimation". ther, Rodrigues

and Zellner (1994) proposed

Fur-

the weighted balanced

loss function, and apply it to estimating the exponential mean time to failure. They obtained optimal point estimators and showed these estimators are a compromise

between usual Bayesian and non-Bayesian

estimators. Various other authors have used this form of loss function in a number

of recent studies including Chung

al. (1999), Sanjari Farsipour and Asgharzadeh

et al. (1998), Dey et (2004) and Gruber

(2004). In Section 2, we discuss alternative loss functions and obtain Bayes estimators relative to them.

We also compute the risk and

Bayes risk functions of c X + d. In Section 3, we investigate the admissibility of linear estimators of the form cX + d. Finally, in section

4, the class of inadmissible linear estimators of the form cX + d is

123

identified.

2

Loss

Functions

and

Bayes

Estimators

On the basis of a random sample X 1 , . . . , X~ from (1.1), one BLF for the present problem is [see Rodrigues and Zellner (1994)] ~t

h(~,0)

= ~

( ~ _ ~)2 + (1 - c o ) ( ~ - 0) 2,

(2.1)

and a particular W B L F is L2(0, 0) = co E i ~ l ( X i - 0) 2

g2

( 0 - 0) 2 +(s-co)~,

(2.2)

where 0 < co _< 1 and 0" is some estimate of 0. In (2.1) and (2.2) the first term oll the r.h.s, represents term represents estimation For a scale parameter

goodness

of fit while the second

precision. estimation,

the commonly

used quadratic

loss is given by

L~(~, e) = (~ - 1) 2.

(2.3)

From this viewpoint, for estimation of the scale parameter 0 in the exponential model, it is helpful to use a W B L F L4 which is given by

L4(~,0)

o2

+ (1 - co)(

- 1) 2.

(2.4)

In (2.4) the first term on the r.h.s, is a quadratic measure of goodness of fit, weighted by co~nO2 and the second term a squared error measure of precision weighted by (1 - co)/02. Note t h a t if co = 0, the loss function (2.4) reduces to the weighted squared error loss

(2.3)

while

if co = 1, the loss function reduces to a pure goodness of fit criterion. Since the B L F in (2.1) and the W B L F in (2.2) have been analysed in Zellner (1994) and Rodrigues and Zellner (1994), we shall focus attention on the analysis of the W B L F in (2.4).

124

For later use, in this section, we first consider the Bayes estimation of 0. P r o p o s i t i o n 2.1. The unique Bayes estimator of 0, say 0B, under the loss (2.4) is given by

E(O-1IX)

(2.5)

E(0-21x ) '

where X = (X1, ..., X~). P r o o f : Since the loss in (2.4) is strictly convex, the unique Bayes is obtained from the relation

OE[L4(O, 0)IX] 0~)

= 0,

which reduces to (2.5). Let A = ~, the conjugate family of prior distributions for A is the family of Gamma distributions, F(c~,/3), with density 71"(/\]Og,/3) =

---~c~--le--Jg~, .,~>

0

(2.6)

where c~ > 0 and/3 > 0. Note that the limiting case c~,/3 --+ 0 gives the usual noninformative prior for A, 7c(A) o( ~-1 (e.g. Berger, 1985 P. 114). It is easy to verify that the posterior distribution of t is F(n +

a, [3+ ~'~-1 xi), and from Proposition 2.1 the unique Bayes estimator of 0 is given by

&

_

,E(e-~rx)

=

wX + ( 1 - w ] ~ )

:

, E(AIX) ~X + ( 1 - ~ j ~ )

=

wX+ ( l - w ) /3 + Ei~=l Xi n+a+l

Note that OB can be written as v~a~ - ~ + (~ + 1 ) ~ n+a+l

+

(1

-

w)/3

n+a+l'

(2.7)

125

which is of the form c X + d. Also note t h a t a~ < -

Remark

-

n+(c~+l)w

n+~+l

n+w

< n+l"

2.1. For the noninformative prior ~(A) o( A-1, the posterior

distribution of A is F(n, ~in__l xi) and we obtain the generalized Bayes estimator

& - ~+~.__ n+l T h e following proposition gives the risk function and Bayes risk of the linear estimator of cX + d. Since, the derivations are straightforward they are omitted.

Proposition 2.2. T h e risk function of the estimator c X ÷ d, relative to the W B L F (2.4) is

R(< c ~ + d)

=

[(c - 1)0 + d]2 e2 + ![(c~ - ~)2 + ~(~ _ ~)], d2

-

2 ( c - l)d

05 +

o

)2

+(c-1

+![(c~ - ~ ) 2 + ~ ( ~ - ~ ) ] ,

and the Bayes risk of c X + d, relative to the prior (2.6) is

r(7~,c-X + d)

3

Admissibility

2(c-

=

d2 c ~ ( ~ : 1) +

1 ) d ~c~ +

+

! [ ( c - ~)2 + ~ ( ~ _ ~)].

(c-

1) 2

n

of cX + d

In this section, we consider the question of admissibility of the linear estimator cX ÷ d. Theorem

3.1. Under

the loss function (2.4), the estimator cX + d

is admissible whenever w < c < n+~ andd>0. P r o o f : Take w < c < n+~ and d > 0. Define

,

= n+~-(n+l)c, C - - ~M

/3*=

nd C--

~d"

T h e conditions co < c < n+~ and d > 0 ensure t h a t cF > 0 and

/3*>0.

126

From (2.7), since = c,

n+a*+l

n+a*+l

-- d,

it follows t h a t cX + d is the unique Bayes estimator of ~ relative to the prior distribution F(a*,/3*). This proves t h a t cX + d is admissible whenw 0

convex subset of (0, oc), it can be shown

there exists a k0 such that fAfck(A)dA

that

> e for some e > 0 and all

k>_ko. The

formal Bayes estimator with respect to 7rk under the loss (2.4)

can be derived as in (2.7) which is given as

nk+k+l

nk+k+l

By proposition (2.2), the risk function of 0k is

mo,£)

(k + 1)2(1 - ~)2 (1 - ~)2k2/32 23k(k + 1)(1 - ~)2 + (nk + k + 1) 2 (nk + k + 1)202 (nk 4- k J- 1)20 ~k2(1_~)2 ~(~_~)

:

+

(~%:i%-U~ +

The Bayes risk of 0k with respect to 7rk is r(7~k, Ok) :

+ Now

(i - w)2k2j32 F(¼ + 2) (nk+k+l) 2 /3¼+2

2/3k(k+ i)(i - w) 2 F(¼ + i) (nk+k+l) 2 /9¼+ 1

[(k +1)2(1- w) 2 nk2(l_m)2 w ( n - w ) F(1) ' ~0. Remark

3.1. The generalized Bayes estimator nn ++l~ "~ is minimax.

To see this, note t h a t the risk function of nn ++l ~

m0, n + n+

~)-(1

is

~)~+ n+l

n

which is a constant number. Since ~+~--Y is admissible with a conn+l "" stant risk function, hence, it is minimax.

4

Inadmissibility

of cX + d

In this section, the class of inadmissible linear estimators is obtained. We

obtain conditions under which linear estimators are inadmissible

128

in terms of risk. The following theorem

gives conditions for inadmis-

sibility. Theorem

The

4.1:

linear estimator

the loss function (2.4) whenever

cX + d is inadmissible

under

one of the following conditions hold:

(i) c < co;

n+co and d > 0; (ii) e > gg-f n+co and d < 0; (iii) co < c < 7g-T (iv) c = c o a n d d < 0 ; (v) c = ~+~ a n d d < 0 .

P r o o f : For c < co, note that (c - 1) 2 > (co - 1) 2 and hence

R(O,c-X+d)

=

~--ff(c-1)210+ c_~dl]2+ l [ ( c - c o ) 2 + c o ( n - c o ) ]

>

0 ~ ( c o - 1 ) 2 [ 0 + c_~d112+ l [ ( c - c o ) 2 + c o ( n - c o)]

>

o~ [(co - 1)0 + d(co 7---i- 1)12" + co(~ - co)

:

R(o, c o x +

--

d ( c o - 1) ct51 )

Thus c X + d is dominated by cox + ~

when condition (i) holds.

For the case (ii), let us compute the risk function of the linear estimator cX + d. From Proposition (2.2), the derivative of the risk with respect to c is --

2d

R ( o , c x + d) = T

+

2[(n + 1 > - ,~ - col ~

(4.1)

Given condition (ii), the equation (4.1) is greater than 0. Moreover, the risk function is a continous and increasing function of c for c > -n+co Therefore, R(O, cX + d) is minimized at co = n+l "

be shown

n-t-CO

n+l

'

Also it can

that

R(O, ~X + d) - R(O, ~ + c o x n+l

2 d [ ( n + l ) c - n -col +d)

=

+

(n + 1)o [(n + 1 ) c - n - co]2 ~(~ + 1) 0,

129

n + w ' - ~ 4_ w h e n condition (ii) holds. Hence c -X- + d is d o m i n a t e d by h--%-i _ d

in this case. For the case (iii), the estimator c X + d is d o m i n a t e d by c X - d. To see this , note t h a t R(0, c X + d) - R(0, c X - d) -- 4(c - 1)d 0 >0, sincec< landd0. For condition (v), consider the difference risk functions of _h_g_fX+dn+w-and nn ++l ~

'

It follows that

R(O, n + w -K n + w-+ d) - R(O, n + 1

+ix) =

d2

+

2(w - 1)d

> O.

Hence ~--~T~ + d is dominated by n+~-~ n+l "~' Remark

4.1. By Theorem 4.1, the sample mean X is not admissible

under the loss (2.4). It is dominated by n+w-~ n+l ~"

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130

[4] Dey, D.K, Ghosh, M. and Strawderman, W. (1999). "On estimation with balanced loss functions". Statistics & Probability Letters.

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