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On time-sequential point estimation of the mean of anexponential distributionPranab Kumar Sen aa University of North Carolina , Chapel HillPublished online: 27 Jun 2007.

To cite this article: Pranab Kumar Sen (1980) On time-sequential point estimation of the mean of an exponentialdistribution, Communications in Statistics - Theory and Methods, 9:1, 27-38, DOI: 10.1080/03610928008827856

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ON TE\:.IE-SEQUEPITIXL P C I M ESTI:*lATIC?I OF TiE :.!EM OF AN EXPONE1;TIAL 9ISTRXBUTIC!!

Pranab Kmar Sen

L'niversity of Xorth Carolina, Chapel M i l l

Key Words and Phrases: a s y m p t o t i c n o r m a l i t y ; a s y m p t o t i c r i s k - e f f i c i e n c y ; l o s s f u n c t i o n ; risk f u n c t i o n ; s t o p p i n g number: s t o p p i n g t i m e ; t i m e - s e q u e n t i a l p r o c e d u r e ; t o t a l l i f e under test.

ABSTRACT

In the context of a l i fe t e s t i n g problem, a t ine-sequential

procedure f o r estimating the mean of an exponential d i s t r ibu t ion

is p r o ~ o s e d and is shown ts be asymptotically r i s k - e f f i c i e n t .

Asymptotic proper t ies of the stopping number, stopping time and

the t o t a l l i f e under t e s t i n g a re a lso s tsdied.

Let { X i 211 be a sequence of independent and iden t i ca l ly i'

dis t r ibu ted (i .i . d . l nonnegative random var iables (r . v .) with the

dis t r ibu t ion function (d.f .) Fe(x3 = l -exp(x/9), x 2 [O, =),

where 3 (> 0) is an unknom parmete r . For a (? 1) i'tems under

'IJ> 2esting, the fsiiicres X n,l , - - . , X a r e the orc'er n,n

x a ~ i s t i c s corresponding t o XI,..., X and, from cost and t i n e n

Copyright %) 1980 by Marcel Dekkcr. fnc.

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SEN

considerations, one may cu r t a i l experimentation a t the k-th

f a i l u r e X and estimate 6 by gnk, where n,k

1 k 6 , =r; Vnk and vnk = 1 x ~ , ~ + (n -klXn,k,

i=l

fo r k -1,. . . ,n. Note tha t Vnk is the totd life zmder test 2

upto the k-th f a i z m , EV* =kg and E(V* - k812 =kg , so tha t

2 -1 8 i s unbiased f o r 8 and ~ ( 8 ~ ~ ) = 8 k , f o r k = 1,. . . ,n .

Thus, i f al and a2 be respectively the cost of recruitment (per

individual) and of foltow-up (per uni t of t e s t - l i f e ) , then one may

conceive of the loss incurred i n estimating 8 by gnk as

a ( 6 - B ) ~ + a n + a V Lnk o nk 1 2 nk' (1 r k rn) , (1 . 2)

where the weights a,, al and a2 a r e given posi t ive constants.

The risk i n estimating 8 by $nk is therefore

%&a, 8) = ELnk =k-'a e2 +al" +a2k8, 1 r k s n , 0

(1.33

where = (ao, al, a2) >g. Naturally, given 5 and n, one would

seek t o minimize the r i s k i n (1.5) by a proper choice of k

(1 s k sn ) . However, it is qui te c lear t h a t a s 8 is not

specified, no s ingle value of k minimizes Rnk(a,, 8) f o r a l l

8 >0. Hence, a sequential procedure fo r choosing such a k is

desired and, in view of the nature of the problem, we a re thus

led t o consider a time-sequential procedure.

Motivated by the work of Robbins (1959), S t a r r and Woodroofe

(1972), Woodroofe (1977) and Ghosh and !fukhopadhyay (1979)

[relat ing spec i f ica l ly t o the c lass ica l sequential p i n t e s t i -

mation problems], i n Section 2 , we formulate the proposed time-

sequential procedure and, i n an asymptotic setup (comparable t o

the one i n the c lass ica l sequential point estimation problem),

study i ts various propert ies . Specifically, under appropriate

regular i ty conditions, the asymptotic risk-efficiency of the

proposed procedure and the asymptotic d i s t r ibu t ion theory of the

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stupping time, swpcing number and t h e total l i f e mder t e s t a r e

considered. The de r iva t ions of t h e main r e s u l t s a r e postponed

t o t h e concluding sec t ion .

2. TIME-SE~~TNTIAL POIKT ESTI:.LZTION OF e

Note t h a t by [1 .3) ,

< aao Rnk(s, 3) $ Rnk+, ( 5 , 9 ) according a s k(k+l) ; - . (2.1)

a2 Thw, i f n(n - 1) < 3ao/a2, then Rnk(a, 8) is i i n ?n(l r k s n ) . hence, k = n is an optimal choice, r equ i r ing the experimentation

t o be continued u n t i l a l l t h e f a i l u r e s have occurred. On t h e

o t h e r hand, if n(n -1) zaao/a2, then t h e r e e x i s t s a

kn(=kn(g, 3 ) ) , such t h a t kn < n and Rnk (a, 9) n = m i n f ~ (5 , 8 ) : 1 'k 5"). In t h i s case , one can s top exper i - nk

mentation a t t h e k - th f a i l u r e ( r e s u l t i n g i n savings of time and n cos t of experimentation) and es t imate B by 6 having t h e

nkn ,minimwrr r isk ( f o r a given n and 5) . By (2 . I ) , we have

Let then 0

R, (z, 3) = R (2 , 8) nkn

I t is q u i t e c l e a r t h a t both kn and R: depend on B and hence,

f o r unknown 3, a r e not workable so lu t ions . A t each f a i l u r e , - 1 w e update t h e es t imator of 3 (using t h e sequence gnk = k Vnka

1 .ck n ) and, motivated by ( 2 . 2 ) , consider t h e stopping

munber ?I ( = N (a ) ] by l e t t i n g n n

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The corresponding stopping **he i s Xn,?u' and t h e t o t s : ,ife n

under the t esz upto the s w ~ p i - t - h e X n , is VnN . The - n n

point est irmtor of 6 i s then =N-'V and the r i s k "nNn n nNn ,-. corresponding to BnN i s

n

We a r e primarily interes ted i n comparing the two r i s k s i n

(2.3) and (2.5) and i n studying the asymptotic behavior of Nn, X and 'n% ) when we impose c e r t a i n asymptotic n,Nn

considerations on 2 and n .

In the c l a s s i c a l sequential point estimation theory [v iz . ,

Woodroofe (1977) and Ghosh and i4dkhopadhyay (1979), among o thers ] ,

the loss function is a l i t t l e more s l r n ~ l e r , namely, a2 = O and A ?

Ln =ao(enn - 0 ) - + aln, and the problem i s t o choose n i n such

a way t h a t the corresponding r i s k i s minimized. In t h i s context,

one l e t s a1 +0 and some optimality r e s u l t s a re val id i n t h i s

asymptotic sense. In our problen, however, f o r a given n and a_. the stopping number Nn depends on a and a2, but not on a

0 1' rhough the r i s k s i n (2.3) and (2.5) depend on a l l the th ree

ao, a l , a,. Here, we l e t a2/ao +O, o r , s i q l y , a2 +O, keeping - a. f ixed. We may note t h a t by ( l .3 ) , f o r every n ' r n ,

Rn,,(g, 0) = Rnk(a, 5) + a ( n t -nl ?Rnk(a, 3) . Further, by (2.2), 1

whenever n > (5a /a2) ', f o r small a,, o kn - (6ao/a,) ' and hence,

by (1.3) and (2.31,

0 R.(%, 0 ) - aln +2(aoa263)4 . ( 2 . 6 )

Thus, f o r a given a2, there i s no advantage i n increasing n

indef in i t e ly . Rather, one should t r y t o choose 3 as small as

possible (but greater than (6a0/a2)'). Since, 5 i s not

specif ied, such an optimal value of n may not be ava i lab le .

However, t h i s suggest the r a t e of increase o f n with a2 t O

and we assume tha t the sample s i z e n(= n(a2)) depends on a2

i n such a way tha t

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TI=-SEQUENTIAL POINT ESTIMATION

l i m a,[n(a,) l2 = a* e x i s t s , f o r some 0 ca* c-. (2.7) a,SO A

- i

Our r e a l i n t e r e s t l i e s i n the case where

when kn w i l l be s t r i c t l y l e s s than n. In f a c t , by (2.2) and

l i m (n-'kn) = y V 0 < y '1. a $0 2

Fiowever, t h e main r e s u l t s t o follow remain t r u e even when y 21, though, i n t h a t case, k = n and ( 2 . 9 ) may not hold. F ina l ly , n a s i n t h e case of t h e c l a s s i c a l sequent ia l point es t imat ion

~ r o b l e m , we assume t h a t a l -LO. Spec i f i ca l ly , we l e t

al = pa2, f o r some (fixed) p E (0, m) . (2.10)

Then, we have t h e following

,"F,eorem 2 . iinder (2 . 7 ) , as a, S 0,

Nn/kn '1 almost surely ( a . s .), (2.11)

and, for every reaZ x ,

where kn sat is j ies (2.9) for y < l ond is s themise zqual ;o

n and $(XI 5s she standard ndmal i. f. Also, (2.13) .nay be

rep laced by

:Ve m y remark t h a t f o r Theorems 1 and 2 , we do not need

( 2 . 3 ) . Also, (2 .15) es tab l i shes the asymptotic equivalence of

t h e two r i s k s R: and R: and, in t h i s sense , t h e yoposed time- Dow

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sequentlal procedure is a s p ~ t o ~ i c a ~ & risk-e f . f - Ic ient . For

(2.15) to hold, in (2.10j, we may even allow o to be arbitrariip

close to 0. Also, if a,/a, +m with a, YO, then both R: A -

and R* are dominated by the common term aln, and hence, a

(2 .lS; holds trlvlally. Thus, (2.10) is not really a stringent

requirement for (2 .IS] to hold.

The assumption (2.8) [or its negazion] has a greater impact

on the asymptotic behavior of the stopping number Kn anl the

stopping time X We have the following n,Sn' ?r ,nearer Z . limier (2.7) a*L ( 2 . S) , cs a2 10,

p{2k-'(N -!i ) 2x1 +@(x ) , Y - m < X cm. n n n

(2.16)

i i i so , ur&r (2 .?) an& f i r a* < @ao,

P{z~'>'(s~ -n) 5x2 -+ (2.18) 11, x 20.

(Xote that N rn, with probability 1, and hence, - 3 n n (Nn -n) is a non-positive r.v.)

,Theorem 4 . limier (2.7), for a* >8ao, a s a2 10,

;ti?i,e, for a* <Ba for every reaZ x, 0'

We may remark that for a* =Baa, we have not been able to

obtain the asymptbtic distribution of XnjK in a simple,

closed form and this we pose as an open prohem. The main

diffuculty in this case stems from the fact that for the simple L

exponential d.f., for any c >O, { x ~ , ~ - ~ + ~ , i r k r cn '1 does

not satisfy the criterion of ufiifomn c o n t i n u i q i n probabiii ty,

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so t h a t X n,N nay not be approximable ( i n p r o b a b i l i t y ) by any n

s i n g l e Y, n,n-k+l ( f o r some 1 < k rcn-'), while t h e n i x t u r e s o f

t hese extreme order s t a t i s t i c s f a i l t o have any simple d.?.

3 . PROOFS OF THE T F I E O E S

::ate t h a t by ( 2 . 2 ) and (:.i),

l i m = +- a . s . n a, +O - For every n ( > 1) , l e t VnO = 0 and

where X = O . Vote t h a t f o r every n ( r 1 ) , Z n l , . . . , Z a r e n,O nn

i . i . d . r . v . each having the d . f . Fq(x) = 1 -exp(x/$), but f o r

d i f f e r e n t n, t h e Z a r e d i f f e r e n t . Also, note t h a t f o r nk

every n ( > 1 ) , f Vnk is I i n k (1 s k s n ) . (5.3)

Fur ther , no te t h a t f o r every q >0,

P{X , f o r some in r n j m , l

- 0, a s n + - . Thus, by (3.2) and (5 .4 ) , f o r every ~ > 0 ,

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3 4 SEN

5 1

a.n "(n5/' + l ) / a o - a -l a*n-lI8. Hence, by (3 .5) , we ob ta in t h a t 5

N > n '' a . s . ( a s a, S 01, so t h a t n -

l i m Nn = + m a . s , a 40

2

Note t h a t f o r t h e t r i a n g u l a r a r r a y <Vnk, 1 2k s n ; n 211, t h e

Monotone Convergence Theorem m a y n ~ t be app l i ab le t o prove (3.6).

Since t h e Bk i n (3.2) a r e i . i . d . and have f i n i t e moments of

a l l f i n i t e order : using the e igh th order noment of t h e V nk' t h e Xarkov inequa l i ty and t h e Bore l -Cante l l i Lemma, it follows

t h a t 1 max Ik- Vnk -81 - 0 a . s . , a s n +m, 5

(3 .7)

n ''-<ka

so t h a t on not ing t h a t H >n' a . s . a s a, + 0, we ob ta in f r o m n

(3.7) t h a t

and i n (3 .8 ) , En m y a l s o be replaced by (N - I ) . n

Let us now s t a r t with t h e proof of (2.11). F i r s t , we

cons ider t h e case when (2.8) holds , so t ~ t kn < n . Then, by

(2.21,

kn(kn-1) <3ao/a2 <kn(kn + I ) . (3.9)

On t h e o the r hand, by ( 2 . 4 )

, r 2 V n ~ -1 > N n W n -1) a2/ao, (3.1C:

n

so t h a t by (3.9) and (3. l o ) ,

By (3.8) and (3.111, we ob ta in t h a t under (2.8),

l i m sup(%/kn)' s 1. a,LO 6

[For kn = n , (3.12) follows t r i v i a l l y by not ing t h a t Nn r n , 7

with p r o b a b i l i t y 1 .] Note t h a t [Nn = n ] = [Lrnk >k'(k + l ) a 2 / a o . 2 -1 -1

'f k < n -11 c [V > (n -1) na,/ao] = [(n - 1) 6 Vnnml > nn-1 - Dow

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TI=-SEQUENTIAL POINT ESTLHATION 3 5

n(n - l ) a2 /3ao] , uhere by (2.7) and (2.5), l i n ~ ~ ~ i n ( a - l l a ~ / a a ~ ; i >1. a,yO

Tks, under 2.7') and (?.9), - N < n a . s . (as a7 J. 0) , where f o r Xn < n, by (1.3) , n -

n

so t h a t b y (3.9) and (3 .13) , f o r !J < n , n

From (3.3) and (3 . l 4 ) , we have under (2.7) and (2.8),

2 l i m i n f (?Jn/kn) > 1,

a +O 2

so t h a t (3.12) and (3.15) in su re (2.11). Xote t h a t when (2.8)

does not hold ( i . e . , a* seao , then f o r every ~ ( 0 < E < I ) ,

k(k +l )a2/8ao r ; l -E, Y k c n ( 1 - E), so t h a t by using (2.4) and - -

(3 .8) , we conclude t h a t N z n ( l - E ) a . s . , a s a2 + O . This n in su res (2.11), when (2.5) does not hold (where k = a ) .

-1" SOW TI- '>I~ 51, with p robab i i i t y 1, while n k is

bounded away from 0 , by (2.2) and ( 2 . 7 ) . Hence, Nn/k, is a

bounded r . v . , so t h a t (2.11) insures (1.12).

- k 9 ) / I 4% 9) is asymptot ica l ly nornal , !lore t h a t (Vnkn ? (2.11) insures t h a t Yn/kn - 1 , while t h e Znk have f i n i t e

moments of a l l f i n i t e o rde r . Hence, by an appeal t o t h e

hscombe (1952) theorem, we conclude t h a t (2 . l J ) holds . Further,

so t h a t by (2.11) , (2.14), (5.16) and the Slutzky theorem, (2.13)

fo l lows. This completes the proof of Theorem 1.

For Theorem 2, we no te t h a t by (2.7) and (2. l o ) ,

(where Oe means an exact o r d e r ) , so t h a t by (t.b),

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3 6 SEN

Also, f o r every n ( > 1 ) , 2 1 < k 5n a r e i . i . d . r . v . with :he nk ' s i x z l e exponential d . f . , whi le , by definition, Y s n , with n ? r o b a b i l i t y i . :ience, we may modify t h e ?roof of Theorem 3 .2 of

IGoodroofe ( 1 9 7 ) and thereby, &voiding t h e d e t a i l s , ob t a in t h a t

Then, (3.15) fo l lows d i r e c t l y from (5.18) and (3 .19 ) . Q.E.D

Let us proceed now t o t h e proof of meorem 3. Note t h a t

under i?.,) and ( 2 . 8 ) , h' c n a . s . , a s a, Y O , and, f o r n n -

?: < n , by (5.14) and t h e d e f i n i t i o n of Snk, n

S imi l a r ly , by (3 . l ? ) ,

where by (2.11) and (3 . ?7 ) , a s a 2 J 0 ,

k-% + O and (Xn + k )/21 -1 a . s . (3.22) n n n

By (2.13) (5 .10) , (3 .21) , (3.22) and t h e Slut:!:? Theorem,

(2.16) fo l lows .

i a e n a* = $ao, k = n , f o r every ?; < n , (3.10) holds n n and (3.21) holds f o r every ?I' s n . Thus, a s i n above, f o r every

n x < 0, ~ ~ 2 n - ~ ' $ - n ) 5 x l - + G(x) . On t h e o t h e r hand, N s n , n n wi th ? r o b a b i l i t y 1, so t h a t ~ ( 2 n - ~ ' ( ~ ~ -n ) 2 0 1 = 1 , b' n and t h i s

proves (2.18 J . To prove (2. I ? ) , we may no te t h a t under (2.7) ,

[a* < Sa,] => [n(n - l ) a 2 / (aoS) -+ a*/ (9a ) < 11 . (3 .23) - 0

Xlen, by ( 2 . 3 ) , (5.83 and (3 .23 ) , i t fo l lows :hat

F i n a l l y , l e t us cons ider t h e proof of Theorem 4 . Under

(2.7) and (2.81, by (2.111, q / ( n y ) P 1 where 0 c y <I. In

f a c r , 5y (?.16), iL' -n? =O- [n") . Let be def ined by n p

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TI=-SEQUENTIAL F'OIINT ESTIMATION

F a ( < ) = 2, 0 < a ( 1 ( i . e . , = - 3 l o g ( 1 - 2 0 , 0 < a c l ) . n l en , v Z Y

by an appeal t o t h e Bahadur (1966) r ep re sen ta t i on of sample

q u a n t i l e s , we ob ta in from t h e above t h a t

while, n- lk Y i n su re s t h a t

(2.29) fo l lows than from (3.25) and (3.26) . To ?rove ( ? . 2 0 ) , we make use o f (3.24) and the f a c t t h a t

P ( ( . ( ~ > ~ - 3 l o g r . ) / i r d - ex?(-e-'!, J r e a l x . Q.E.D.

This work was supported by the Yational Hear t , Lung and

Blood I n s t i t u t e , Cont rac t Xo. NIH-NHLBI-71-2243 from the !:ational

I n s t i t u t e s of Health. Thanks a r e due t o t h e r e f e r e e f o r h i s

very use fu l comments on t h e manuscript . These have l ed t o t h e

e l imina t ion of t h e proof of Theorem 2 and s i m p l i f i c a t i o n s e l s e -

where.

.bscombe, F . J . (1952) . Large sample theory of s equen t i a l e s t ima t ion . MOC. Cmbr*:@e ?$i. Soc. 58, 600-7.

Bahadur, R . X . (1966). A note on q u a n t i l e s i n l a r g e samples. Ann. ' b z h . Stn~isr. 57, 5 7 7 - 8 0 .

C ~ O W , Y .S . (1960) . A s a r t i n g a l e i n e q u a l i t y and tile law of l a r g e numbers. ,>-roc. &er. .'dcth. Soc. LA, 107-11.

Chow, Y . S . , Robbins, H . and Teicher , H. (1965). :.lonents of randomly stopped sums. Ann. :4azh. Statiss. 35, 799-99.

Ghosh, :.I. and :lukho?adhyay, ? I . (1979). Sequent ia l 7 o i n t e s t i n a t r o n of t h e aean when t h e d i s t r i b u t i o n i s m s p e c l f i e d . :om. 3tctLx. AS, 637-52.

Rosbins, 9. (1959). Sequent ia l e s t i n a t i o n of t h e Dean of a normal ~ o p u l a t i o n . h.sba2i't ~3 xi S5ccCs~ics (H. Cram& vo17me), Xlmquist E Wiksell, i ippsaia, 235-45. D

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S t a r r , N . and Woodroofe, b l . ( 1 9 7 2 ) . Further remarks on sequent ia l esti iaation : the exponential case. A m . Xzth. Z t n t i s z . 4.3, 1147-53.

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R e f e r e e d b y M a l a y G h o s h , Iowa S t a t e U n i v e r s i t y , A m s , I A .

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