Unified jackknife estimation for parameter changes in an exponential distribution

UNIFIED JACKKNIFE ESTIMATION FOR PARAMETER CHANGES IN AN EXPONENTIAL DISTRIBUTION

JUNGSOO Woo

1. Introduction

Many authors have utilized an exponential distribution because of its wide applicability in reliability engineering and statistical inferences (see Bain & Engelhart(1987) and Saunders & Mann(1985)). Here we are considering the parametric estmation in an exponential distribution when its scale and location parameters are linear functions of a known exposure level t, which often occurs in the engineering and physical

phenomena. .

The purpose of this work is to estimate the effects on the scale and location parameters in the exponential distribution when both parameters change a function of environmental dosage, say t. First, we assume an exponential model and estimate the parameters based upon the complete or truncated samples by the maximum likelihood and jackknife methods. The derived estimators will be shown to be asymptotically unbiased and mean square error(MSE)-consistent under a niced condition.

Throughout the numerical evaluations of biases and MSE's of the maximum likelihood estimators and its jackknife estimators for the scale and location parameters in the small sample sizes, the biases and efficiencies of the proposed estimators will be compared each other.

Received October 4, 1993.

Key words: complete and truncated random sample, MLE and truncated MLE, Jackknife estimator, MSE-consistent.

Research was supported by Yeungnam University Research Fund., 1993.

2. One parameter exponential distribution

We are considering an exponential distribution with the pdf f(x; u(t)) =u~t) ^{exp {-}u~t)}' x> 0, u(t) > 0, written by X '" EXP(u(t))o

Here we are considering a unified estimation for the parameter chan- ge of exposure levels or times in one parameter exponential distribution even when the parameter is a polynomial of ^tj

u(t)=bo+blot+ooo+brotr,. t>O, bi >0, i=O,l,ooo ,ro

2.1 The complete samples

AssumeXli,⁰o. ,Xn'i be a simple random sample(SRS) taken from_{J .} Xi '" EXP(u(ti)), j = 1, ^{0 0 0} ,r+1, andXl,^{00 0} ,X_{r H} be independent,

ti -#~tk"fori ¥ k, -. .

Define the following notation:

1 tl t~

tr

1 t2 _t~ t2

det [t~,tL^{0 00 ,tn=}

1 tr+l t;+l t~H

By the maximum likelihood method, we can obtain the maximum likelihood estimators(MLE) for bi;

d t [to ti-l X ti H t r)

~(l) e · ^{0 00 • . ' . 0 0 0 •}

b - ^{1 , ' I ' . . 1 ' 1 , ' I • -}

°

j - det[tfJ ⁰ o. tr:) , J - " "

I ' , 1

where X.i = _{~i L~~l}Xki, i = 1, ^{0 0 0} ,r+1.

The expectations and variances of these estimators b)l) j = 0,.⁰⁰,r, are given by

E(bJl») =b_j and

r+l 2(t)d t²[to ti -¹ ti+ l t r]

~(l)

L

VAR(b. )=

J nkdet2 [tfJ ⁰⁰⁰ tr:) ,

k=l I ' , 1

h d t [to ti-l t^j+l r) ^{0 0} determinant eli ⁰

were e i,^{0 0 0 ,} i , i , ..⁰ ,t_i iy!k IS a mInor eterminant e ^lll1-

nated k-row and j-column in the determinant, det [t1,^{0 00} ,tn.

Therefore, we get the following.

PROPOSTION 1. Tbe MLE's b)l),j = 0,. ..,r, are unbiasedandMSE- consistent estimators ofb_{j ,}respectively.

2.2 The truncated samples

For giventi =1=tkfori =1=k,1,. .. ,r+1, letXlj, ... ,Xkjj, ... ,Xnj^{i be} the truncated random sample(TRS) taken from Xj '" EXP(a(tj», and Xl,··· "Xr+l be independent, whereXlj,··· ,Xkjj are dead items or item of failures and Xkj+lj, ... , Xnjj are alive items or runouts, j = 1,.··, r+1, and

The likelihood functions are given by

k· n'

J 1 Xij ^J

L(bo, · · · .b;Itj) =

II

II

i=l J J i=kj +1 { Xi^{j }}

exp - a(tj) ,

and

Ifwe assume the truncated number Kj - 1 follows a Poisson distri- bution with mean Aj and Kj's are independent, j = 1,··· ,r+1, then the expectations and variances of b}2) ,^{j =} 0,.·· ,r, are given by

E(W»)

:E~-o bm^det [t~,... ,t{-l,^{(1 - exp}(->'i))nitr/>.{+l ,t{+\ ... ,tr]

det[t?, ... ,til

VAR(b~2»)

1

:E;t"';l^det [t~,...,t{-\ t{+l, ... ,til i#m det [t~,... ,til

{nm(n m+1)A(>'m;km) - n;'"^{(1 - exp}(->'m))2/>.;"}u 2(tm), where A(Am;km)⁼ L::'oAm exp (-Am)/((X+ l)(x+ I)!).

Therefore, we get the following.

PROPOSITION 2. Ifevery truncated number Kj - 1 follows a Poisson distribution with sufficient large meanAi andK_j 's areindependent,j = 1,. .. ,r+1, then the MLE's b;2) ,^{j =} 0,.· . ,r are asymptotically unbiased andMSE-consistent estimators of bj, respectively.

3. Two parameter exponential distribution

Here we are considering two parameter exponential distribution with the pdf

1 . x-pet»~

I(x;O'(t),pet»~ = O'(t) exp ( - O'(t) ,x > pet), O'(t) > 0, written by x'" EXP( O'(t),p(t) ).

We are considering a unified jackknife estimation for the parameter change exposure levels or times in. two parameter exponential distribu- tion even when two parameters are polynomials of t.

O'(t) =bo+^bi . t+ +^br . tr, pet) =ao+al .t+ +ar· tr, where t >0, ai > 0, b, >0, i = 0, 1, ... ,r.

3.1 The complete samples

3.l.A The maximum likelihood method

AssumeXlj,'" ,Xnjjbe a SRS taken fromx, '"EXP(O'(tj), p(tj», j = 1, "', r-l-L,and Xl,'" ,Xr+l be independent, ti =I tk for every i =Ik.

By the maximum likelihood method, we can obtain the MLE's ofaj and bj,j = 0,1,.·· ,r;

(3) det[t~ ... t~-l X(l)' t~+l ... t":]

a. = J, ' J ' " J ' 'J and

J det[t~_{J ,}... _{' J}if]

b·~(3)_J det[t~_{J ,}... 'It~-l^{, . J}X' - X(l)'" I "t~+l ... t":]I

det[t~_{J ,}... _{' J}t":]

where X(l)j, j = 1,.·· ,r+l, is the smallest order statistic amongXlj, .,. ,Xnjj.

The expectations and variances of these MLE's 0,;3) and bJ3), j - 0,1,. .. ,r, are given by

E[a

tn

= aj +L...i k _det[_{t· ... t·}0 r] ,

k=o a' , a

Therefore, by taking limits for those expressions.

PROPOSITION 3. Th.eMLE's0,;3) andb;3),j = 0,1" ..,r, are asymp- totically unbiased and MSE-consistent estimators for aj andbj,respec- tively.

3.1.B The jackknife method

By definition of the jackknife method, we can obtain the jackknife estimators of 0,;3) and b;3), for everyj = 0,1, ... ,f,

J(a<3) =n.a~3)_ (n. _ 1)a~3)-1

J J J

_ det [t?, ... ,t{-l,«2n. -1)X(1)i - (n. -1)X(2)i),t{+t,· .. ,til

- det [t?, ... ,til ^and

J(b<3) =n.b~3) _ (n. _ 1)b~3)-i

J J J

_ det[t?, .. , ,t{-1,(n.X.i - (2n. - l)X(l)i+(n. - 1)X(2)i) ' t{+1, ... ,trJ

- n. det _[t~,... ,til

where n. =nl +n2+ ... +nr+l.

Also, the expectations and variances of these jackknife estimators can be obtained by, for everyj = 0,1, ... ,r,

[J(^{~(3» )] 1 ~b d [0 i-I}

E aj =aj - [0 r] L..J k et ti , · · · ,t_i ,

n.det t_{i , · · ·} ,t_i k=o

(n. - _ni)t~/ {ni(ni - 1)},t{+I, ...

,tn,

1 ^r

E[J(b(.3»)] = .

L

J n. det [t~_{I '}... _{' I}f':] _k=O

{n.(n~ - ni +1) - ndt~/ {ni(ni -1)},t{+I, ...

,tn

Therefore, by taking limits,

PROPOSITION 4. The jackknife estimators J(o'J³»)and J(bJ³»),j = 0,. .. ,r, areasymptotically unbiased and MSE-consistent estimators for aj and bj, respectively.

Biases and mean square errors(MSE) of the ML estimators and their jackknife estimators for aj 's and bj's in the small complete samples can

be numerically evaluated.

3.2 The truncated samples

For giventi =1=tk for i =1=k, 1,2,.·· ,r+1, let Xlj,··· ,Xk;j,··· ,Xnj;

be the theTRS taken fromXj '"EXP(u(tj), /-l(tj )),andXli,"· ,Xkjj are dead items or items of failures andXkj+lj, ... ,Xnjjare alive items or runouts, j =1, ... ,r + 1, and Xl,'" ,Xr+l be independnet.

The likelihood functions are given by L(a, bItj)

k· n·

II

J 1 x., - ^/-l(t·)

II

_ - - e x p - J J

- . u(t .) { u(t .) }.

1=1 J J I=kj+l

and hence, the MLE'sa~4) ^and b~4)^{for aj andbj,j =} 0,.·· ,r, are given by

A(4) det[t?, . . . ,tf-l, X(l)i, tf+!, ... ,tn

aj det[t0 ]_{i , · · ·} ,tf and

A(4) det[t?, ... ,tf-l,(X.i-

X(1)i)/ki,t1+!,··· ,tn b·J det[t~_I'... t':]_{' I}

Ifwe assume the truncated number Kj -1 follows a Poisson distri- bution with mean Aj, j = 1, ... ,r + 1 and K;'s are independent, then the expectations and variances of a)4) ,j = 0,1,'" ,r,are the same as those of a~3), because the a~3) ^and a~4) are equal and the expectations and variances ofb;4) are given by, for j = 0,1, ... ,r,

E(W)

Ek-Obk det [t~, ... , t{-1,(ni - 1)(1 - exp(-Ai»)t; P'i,t{+1, ... ,

tn

det [t~,···,

tn

VAR(b~4)

J

1 ^r+l

det₂[tQ ... t_f^] L ^u2(t m ){ nm(nm - l)A(A m;km )

.. ' ' .. m=l

- (nm - 1)2 (1 - exp(-Am»)2fA;'} det 2 ft~,...,t{-1,t{H, ... ,ti1#,

where A(Am;km)⁼ 2::'0A~exp(-Am)/((X+ l)(x+ I)!).

From the expectations and variances, we get the following.

PROPOSITION 5. Ifevery truncated number Kj -1follows a Poisson distribution with sufficient large mean Ai and Kj's are independent,j

=^1,.··,r, then the MLE's a~4) ^and b~4) ^{ofaj and bj, j =} 0,.··,r, are asymptotically unbiased and MSE-consistent, respectively.

Biases and MSE's for the truncated ML estimator for the scale pa- rameter in small truncated samples can also numerically evaluated.

Throughout the exact numerical evaluations of biases and MSE's, the jackknife technique is very useful in the bias reduction, but MLE's are more efficient in the small samples than truncated ML-estimators and the jackknife estimators.

References

1. Lee J. Bain and Max Engelhart, PWS Publisher (1987), Introduction to Prob-

~bility and Mathematical Statistics.

2. S. Kim, Estimation of Changing Parameters Based on the Truncated Samples, M. S Thesis, Yeungnam University, 1989.

3. Sam C. Saunders and Nancy R. Mann, Workshop on Statistical Reliability for Engineers and Scientists, National Bureau of Standard, U. S. A., 1985.

4. Sam C. Saunders and J. Woo, An Application of Weibull Distribution to the Strenght of Dental Materials, Technical Report, WSU, Pullman, U. S. A., 1989.

5. K. Yoon, Unified Estimation for Parameter Changes in Two Parameters Ex- ponential Distribution, M. S Thesis, Yeungnam University, 1989.

Department of Statistics Yeungnam University Kyongsan 712 - 949, Korea