Estimation of One Parameter Exponential Family Under A Precautionary Loss Function Based on Record Values

A great deal of research has been done on estimating the parameters of these distributions using both classical and Bayesian techniques, and a very good summary of this work can be found in Johnson, Kotz, and Balakrishnan (1994). Record statistics is first studied by Chandler (1952) and is widely used in statistical modelling and inference such as in several real-life problems involving data relating to weather, sports, economics, and life tests.And it has now spread in different directions. There are also some papers on estimation and prediction for parameters of some life distributions based on record values.See for example Balasubramanian and Balakrishnan (1992), Jaheen (2004), and Ahmadi, Doostparast, and Parsian (2005) and references therein.

X.X

Let 1 ,  2 , be a sequence of independent and identically distributed (iid) random variables with cumulative distribution function (cdf) F(x) and probability density function (pdf) f (x) . Forn ≥ 1 , define

* Corresponding author.

.

{X,„_.}

The sequence ^{1   U ( n} )   is known as upper record Values.

In this paper, we consider the estimation of a special one-parameter exponential family based on record values. Bayes and empirical Bayes estimators are obtained in Section 2. The estimators obtained are special cases of the more general class of linear estimators of the form (cT + d)

-1

. In Section 3, the admissibility and

inadmissibility of ^{( cT +} d ) in Section 4.

are studied. Conclusions and a brief summary of the results are finally provided

Let 1,  2, be a sequence of nd random variables from the class C of one-parameter exponential family with cdf

and pdf

where ⁰ >  ⁰ and a ^{( ), ( x} ) is a real positive function.

The family in C is well-known in the lifetime experiments which includes several well-known lifetime distributions such as: Exponential, Pareto, Lomax, Burr type XII, Resnick, Weibull (one parameter) among others.

2. Bayes and Empirical Bayes Estimation

.  .   X,„ = x x,X,,, = x x..

Suppose we observe n upper record values ^{U (1)    1        U ( n} )     ⁿ from distribution cdf (1) and pdf (2).

X„„,.X„, ••-. X,,,.,

Then, the joint distribution of   U(1)    U(2)        U(n) is given (see Arnold et al. (1998)) by n -1

n i=1

1     ⁰ <  x <  x O < ' <  xM   1 ji        • 1 1 г г ^XIKп\ ■   ■    1

where       ^{1    2           n} and the marginal pdf of    U ^{( n} ) is given by

f (x -0\=     -0\ R__(xn ;0)

Jn (xn ;0 )   J (xn ;0 )   /

H (x^ = where X  (x1,   , xn) , H() is the hazard function corresponding to pdff () ,

f (x t;9)

.

For the later use,we give the following results which can be found their proof in Ren(2010).

X,,, AX = x, , X,„ A = x„ ,— , X,„ X = x

Lemma 2.1 Suppose we observe n upper record values   U(1)    1 U(2)    2       U(n)    n from distribution cdf (1.1) and pdf (1.2).Then

X

1)     ^{U ( n )} is a sufficient statistics for the first n upper record values.

.  .    X,„ = x x ,, —, X,,, = x x..

Lemma 2.2 Suppose we observe n upper record values    ^{u (1)    1        u ( n} )     ⁿ from distribution cdf (1)

and pdf (2) and a ^{( 9 )} = ⁹ ,Then the maximum likelihood estimator (MLE) of ⁹ is _ˆ

A. Bayes Estimation

X.-X. ^-A

In the following discussion, we always suppose ¹ , ² , be a sequence of iid random variables from the class C and ^{a ( 9 )} = ⁹ . And n upper record values U^{u (1)} x ¹ , "^U ⁽²⁾ x ² , , ^^{U ( n} ) x ⁿ are also given. we will consider the Bayesian and empirical Bayesian estimation of θ under a precautionary loss function which proposed by Norstrom(1996), The asymmetric precautionary loss function is

L ( 9 , 9 ) =

(О - 9)2

θ ^ˆ

The loss function approach infinitely near the origin to prevent under estimation, thus giving conservative estimators,especially when low failure rates are being estimated.It is very useful when underestimation may lead to serious consequences.For instance ， in the case of estimation of a financial charge or size of an order,underestimation has much more serious conseququences.

The Bayes estimator under precautionary loss(5) is denoted by θ ^B ,and is given by ^B

Suppose the conjugate family of prior distributions for ⁹ is the family of gamma distributions, ^{Г ( а} , ^{в )} , with density

α

п(9|а, в) = в—0

Г ( а )

'^{a - 1} e ^{- e9} , 9 >  0

. Note that the limiting case ^a , ^в ^ 0 gives the usual noninformative prior

for n(9) ^ 9

estimator of θ ,

.Then the posterior distribution of ⁹ is ^      ^       " ^^ , and the unique Bayes

θˆ say B , is given by

e_B = [ e (6 ²1 x )]^1/2 = [ ^{( n + а )( n + а +} 1) ]^1/2

^B                    ( {+ + T ( X u ( n ) ))²

That is

A

6B = [

^T ( X U ( n ))  _ ₊  ________________

4 ( n + а )( n + а + 1)   4 ( n + а )( n + а + 1)

в

] ^{- 1}

which is of the form         ^{u ( n )}

Remark 2.1. For the noninformative prior ^{n (} ) ^ obtain the generalized Bayes estimator

Г                 -1

¹                                    Г ( n , T ( x„ ))   ,

,the posterior distribution of is           ⁿ and we

л

6 =

T ( X u ( n ) ) 4 n ⁽ n + 1)

B. Empirical Bayes Estimation

is the family of gamma distributions,

Assume that the conjugate family of prior distributions for ^ = ⁶

^{Г ( а , в )} with parameter ^а and ^в ,and prior parameter ^в is unknown. we may use the empirical

Bayes approach to get its estimate. From (3) and (6), we calculate the marginal pdf of

X = (XU  "^' XU(n)), with density fto f (x|6 )п(6|в) de

и

= Г 6 n exp{ - 6 T ( x „)}fj T ( x ,)^- 6^{а - 1} e~ ^ee d 6 = — ^Г^ n ^{+ а} L_J n j T ( _x )

^{Ju     P1 VJ} =1      Г ( а )               Г ( а )( в + T ( x n )) n ^{+а П}= 1 ^{V J}

"                в = ™ t ( x )

Based on m ^{( x 1 в )} , we obtain an estimate ^в of ^в . The MLE of ^в is       ⁿ

Now, by substituting for in the Bayes estimator, we obtain the empirical Bayes estimator as

⁶ EB = ^[

^aT ( X u (n ) ) n

^T ( X u ( n ) )      ₊ __________________

4 ( n + а )( n + а + 1)   4 ( n + а )( n + а + 1)

] ^{- 1}

That is

^{A 6} EB = ^[

n + а

n д/ ( n + а )( n + а + 1)

T ( X u ( n > )Г *

3. Admissibility of ^{( cT + d} )

Note that the estimators obtained in Section 2 are special cases of the more general class of linear estimators of     the

form

,

*

c

.

,

In     this     section     we     always     let

In the rest of this paper, these estimators are compared on the basis of their risks under the loss (2.3). We also obtain conditions under which linear estimators are admissible in terms of risk.

Theorem 3.1. The estimator ^{( cT + d} )

is admissible, provided ⁰ <  ^c <  ^c * and ^d >  ⁰ .

Proof. From (7), when ^a >  ⁰ and ^в >  ⁰ , the coefficient of ^T is strictly between 0 and ^c , and the

J ( n + a )( n + a + 1)

constant

в is strictly bigger than 0. This proves that (cT + d)

is admissible for

is admissible since it is the only

. .   ,   -      . R (0, d) = 0  ,   0 = d estimator for which v ’  7     whenu u

Theorem 3.2 Let the parameter space be ^{(0, +ю} ) and the action space be ^{[0, +ю )} The linear estimator

[ cT + d ]

is inadmissible under the loss function (2.3) whenever one of the following conditions holds:

(i) ^c <  ⁰ or

d < 0. ;

*

(iii) ^c >  ^c and ^d — ⁰

Proof. To see (i), note

takes on negative values with positive probability. Therefore

is dominated by

™⁽⁰,^{( cT + d} )¹ in this case.

( cT) ^-1.

For the case (ii), using lemma 2.1, the risk function of        is

0 c (n —1)

Then derivative of the risk with respect to c is ：

—

1  0 a

2    , < 0

Thus the risk of ^{( cT} )

*

is minimized at c = c

.

Hence the estimator ^{( cT} )

1.                  ( c * T )

is dominated by c

in this case.

For case (iii), let us compute the risk function of the linear estimator ^{( cT + d} )   .

*

R ( в , ( cT + d ) ^{- 1}) - R ( в , ( c * T + — d )) ^{- 1}

c

= в E ( cT + d ) + E (----- cT + d

*

) - в ² E ( c T + — d ) c

- E (-- 1^ )

c *T + — d c

*

= в ²(1--) E ( cT + d ) + (1 - —) E (

c

c

) cT + d

. A ²                     1

= ( c - c *)[ - E ( cT + d ) - - E (

c

c

"FT:?)] cT + d

. A ²                       1        1

> ( c - c *)[ — E ( cT + d ) - - E (— )

*   cT

= ( c - c * )[

c

в2 cn

c

1 в

*

-

c в c * c ( n - 1)

]

> ( c - c *)[ п в- -] c *²( n - 1)

= ( c - c * )[ п в - n ( n - 1)       ] = 0

⁽ n ^- 1)

Therefore, R ^{( в} ,^{( cT + d} ) is minimized at ^c = ^c . Hence ^{( cT + d} ) this case.

1.   .   ■ _t . .    ( c * T + d )

is dominated by

in

A

A

в„. в™

Remark 3.1 Using Theorem 3.3, the estimators ^MLE andb ^EB are inadmissible. They are dominated by the

_t в = ( c * T ) ^{- 1} estimator

.

4. Conclusions

Based on record statistics, this paper considers the estimation of the unknown parameter from the one parameter exponential family. The maximum likelihood estimator, Bayes and empirical Bayes estimators are

obtained. These estimators all belong to the class of inverse linear estimators of the form ^{( cT + d} )

,where

.So, the admissibility and inadmissibility of

are studied. As a result, the empirical

Bayes estimator and the maximum likelihood estimator are inadmissible.