Statistical Papers
Statistical Papers49, 121-131 (2008)
© Springer-Verlag 2008
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.
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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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