Efficiency and superefficiency in one-parameter two ... - Springer Link

BULLETIN

Efficiency and superefficiency in one-parameter two-sided truncated distribution families CHEN Guij ing' and James C. F U ~ 1. Department of Mathematics, Anhui University, Hefei 230039, China; 2. Department of Statistics, University of Manitoba, Winnipeg, R3T 2N2, Canada Keywords: one-parameter two-sided truncated family, superefficiency, a new efficiency, asymptotically median unbiased efficient estimate.

CONSIDER the one-parameter two-sided truncated family: ~ P , ( x ) = f ( ~ 8;) ~ ( Ge G + ( e ) ) d x , e E R, (1) where 8 is an unknown parameter, (Ir(8) is a continuous differentiable function such that 8 < + ( O ) , s ( 8 ) A d + ( 8 ) / d B > O , 8 E R , and f ( x ; 8 ) is a positive continuous density function on [ 8 , (Ir ( 8 ) 1 . The one-parameter two-sided truncated distribution families ( 1 ) possess

Chinese Science Bulletin

Vol .42 No. 21

November 1997

BULLETIN many unusual statistical properties. For example, Chen and chen[ll found the nonexistence of UMVUE of O for any sample size n ; Fu et a l . ['I pointed out that in the uniform family

all MLE of O are consistent, but have different rates of convergence to 8 ; Ibragimov and Has'rninskii ( 1 ~ ) ' constructed ~' a superefficient point estimate for 8 in family ( 2 ) under Bahadur's sense for class of consistent estimates[4*51. IH considered further a density family ~ P ~ ( X = )f ( -~ o l d x , e E R, (3) where f ( x ) > O , for I . z / < l , f ( x ) = O , where ( x I > l . Thedensity f ( x ) isassumed to be sufficiently smooth so that

and (logf ( x ) )"GO, x € [ - 1, 1] . Then they proposed a conjecture: For a MLE

8,

of 8,

1

We check on this conjecture and find out it is not true. In fact, in (3) if we take f ( x ) = -I 2 ( - l < x < l ) , then I = O and ( l o g f ( x ) ) " ~ O , x E[ - 1,1]. Let X1,...,X, be iid. sample, X ( l , = minxi, X(,) = maxXi. Then MLE of 63 are Bnp= p ( X ( , ) - 1 ) + ( 1 - P ) ( X ( ~+) I ) , P lGiCn

IGiQn

E [0, I]. We getL2," 1

limlim-logPo( I - 8 I > y) = (6) 7-0 a-ny where we write a A b = min{a , b 1 . This shows that ( 5 ) does not hold. But the idea about superefficiency of IH is still inspiring. chenL6]has considered two-parameter two-sided truncated family d p O ( x ) = f ( ~ ; 8 e~2 ), 1 ( e 1 G G e2)dx, el e,, (7) where f ( x ; e l , 0,) is a positive continuous density on [ e l , 02] g( 8 ) = g ( 02) is estimated, denoted by ci = a g / a Oi, f i = f ( Oi ; e l , 02), for i = 1, 2. Then under some regular of g , conditions, for each consistent estimate

.


E ) 2(8) r-0 n-rn nE and for the MLE i n , it just reaches the lower bound of ( s ) ' ~ ' . Does there exist a similar result to ( 8 ) in the one-parameter two-sided truncated family ( 1) ? The answer is in the negative. Consider family ( 1) . Suppose g ( 0 ) is to be estimated, which possesses continuous derivative g' ( 8 ) . Denote by dgall the consistent estimates g, of g(B). We have: Theorem 1. Under family ( 1) , for a >0 fixed, there is no function B, ( 8 ) of 8 in R such that B g ( 8 ) > - a and for each

in€4,

1

liminf liminf---logP81 r-0

n-rn

n ~ a

,;I

-

g(O)l

> el>

B g ( 0 ) , foreach 8

E R.

(9)

In fact, for p E [ 0 , 11 and 60E IR fixed, let

g,

+ (1 - ~)X(1,)9 if at least one Xj $ [80, +(80>1, , if all Xj E LOO,$(Oo)I.

= g(p*-l(x(,))

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(10)

November 1997

Then

inE rQ, and PO"I in= g ( Bo) i = 1. This example shows that the phenomena of supereffi-

ciency in any asymptotical efficiency sense cannot be eliminated in family ( 1 ) if such estimators are not removed. For the sake of removing superefficiency in the distribution family ( I ) , we consider the class of AMUE (asymptotically median unbiased estimate) instead of the class of consistent estimate, and consider a new efficiency instead of Bahadur-type efficiency. For any OER, t E R , denote O n = O+ t l n . For an estimate 2, of g ( 8 ) , if (11) !,+.iPem { . ( i n - g ( 0 n ) ) x 1 = G e ( x ) ,


c-0

€

a-m

€1,

1

l * ( O ; g n , g ) = limsup l i m s ~ ~ - l o g ~ ~ i n 1g 8( O~ ) I > e l . o-0

€

n--

(13) (14)

About this type of efficiency we get the following bound in 3,. Theorem 2. Under the one-parameter two-sided truncated family ( 1 ) , for each E %, one has

q =( & i n , g )

,

2- gJ y( O0)) , ,

gn

(15)

w h e y e J ( 8 ) A f l + s f 2 , f l = f ( O ; O ) , f 2 = f ( + ( 8 ) ; 8 ) , S = +'(8). Remark 1 . This theorem removes the superefficiency phenomenon in gg, in family ( 1 ) . But the lower bound does not depend on Fisher information. It could be justified that Ice)

=AZ-, -

+( 0)

lim -

as-o+AO

(f(x;O

J

+(El

+ AO) - f ( z i 8 ) ) d r

(~(x;B + AO) - ~ ( x ; o ) ) ~ x .

(16)

In view of (15), we could say that J(B) is an information function of the distribution family (1) Based on Theorem 2, we shall give the definition of efficient estimate under the distribution family ( 1) . Definition. In family ( I ) , for the estimate gnE a, if is an (asymptotically median unbiased) efficient estimate of g( 8 ) . Does there then we say in exist an efficient estimate in such sense? The answer is in the affirmative. For the parameter function g ( O ) one can construct asymptotically median unbiased estimate as follows: Let f i n

= f ( X ( l ) i X ( l ) ) , f2.n = f ( X ( n ) i X ( l ) ) ,

gn* = g ( ( l - p n Y ) X ( l )+ ~ n * + - ~ ( X ( t z ) ) ) . (18) We obtain the following result. Theorem 3 . I n family ( I ) , the estimate g," belongs to !21g and is asymptotically effiChinese Science Bulletin

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1773

cient estimate in the sense of (17). The estimates g," contain a lot of important and interesting estimates. For example, in 1 1 the uniform family (2), when p = -, the MLE = ( X(,) + X(,) ) is AMUE efficient es2 timate of 6. Remark 2. It could be justified that & g,.

an, I-

in

(Received November 5 , 1996)

References 1 Chen, G. J . , Chen, X. R . , Nonexistence of UMVUE for the parameter of a two-sided truncated family, I . Math. Research & Exposition, 1984, 4(3) : 93. 2 Fu, J . C . , Li, G. , Zhao, L. C . , On large deviation expansion of distribution of MLE and its application, A n n . Inst. Statist. M a t h . , 1993, 45: 477. 3 Ibragimov, I . A . , Has'minskii, R. Z . , Statistical Estimation, Berlin: Springer-Verlag, 1981. 4 Bahadur, R. R . , Rates of convergence of estimates and test statistics, A n n . Math. Statistics, 1967, 38: 303. 5 Bahadur, R. R . , Some limit theorems in statistics, Regional Conference Series in Applied Math, Philadelphia: SIAM, 1971. 6 Chen, G. J., Optimal convergence rates and asymptotic efficiency of point estimators under truncated distribution families, Statistics and Probability Letters, 1996, 30: 321. Acknowledgement This work was partially supported by the National Natural Science Foundation of China (Grant No. 19671001 ), and NSERC of Canada, under grant A-9216.

Chinese Science Bulletin

Vol .42 No. 21

November 1997