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Metrika, Volume 30, 1983, page 49-54.

On Uniformly Minimum Variance Unbiased Estimation w h e n n o C o m p l e t e S u f f i c i e n t Statistics E x i s t

By L. Bondesson, Ume~ 1)

Summary: A simple generalization of the Lehmann-Scheff~theorem is given. It is used to find cases when UMVUE'sexist but complete sufficient statistics do not. Another method to find such cases is also presented.

1. Introduction

The Lehmann-Scheff~ theorem, which in particular asserts that all estimable parametric functions can be UMVU-estimated if a complete sufficient statistic exists, is well known. Less well known is Bahadur's [1957] converse: If every estimable parametric function admits a UMVUE, then a complete sufficient statistic (more correctly: L 2-complete sufficient o-field) exists (provided that the family of probability measures under consideration is dominated by some o-f'mite measure). However, when a complete sufficient statistic is lacking, there may sometimes still be nonconstant parametric functions that can be UMVU-estimated. This fact is very seldom pointed out and exemplified in under-graduate or graduate textbooks. In Rohatgi [1976, p. 356] it is mentioned but the illustrating example is illusory; see Section 6. A correct example is given in Rao [1973, p. 379]. However, it seems as though no example of the phenomenon is well known. In this note there is presented an extremely simple generalization of the LehmannScheff~ theorem which immediately yields many natural examples of cases when UMVUE's exist but complete sufficient statistics do not exist. In spite of the simplicity of this generalization, the present author has not been able to find it explicitly given in the literature. A result by Fraser [1956], which is included as a special case, comes closest. Another simple method to obtain cases when UMVUE's exist but complete sufficient statistics do not is also given in this note.

l) Lennart Bondesson, Department of Mathematical Statistics, University of Ume~, S-901 87 Ume~. 0026-1335/83/010049-5452.50 9 1983 Physica-Verlag, Vienna.

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L. Bondesson

2. General Remarks Although all results below are, like the Lehmann-Scheff~ theorem, valid for an arbitrary convex loss function, they are, for the sake of simplicity, formulated for the squared-error loss function. All estimators we consider are assumed to have finite variances; the qualifying "with fmite variance" is thus always omitted below. A statistic is said to be a UMVUE if it is a UMVUE of its own expected value.

3. Main Result

We assume here that the family of probability measures on the sample space X has the form (Po ,~' 0 E 0, r E ~). A random element in X is denoted by X.

Lemma: Let S (X) be a statistic. If, for every fixed r S (X) is a UMVUE, then S (X) is a UMVUE [sic!].

Proof: We know that if $1 (X, r is a b-dependent statistic that has, for every fixed r the same expected value as S, then var 0,r [S] ~~ 1,N=/:No), where PN ( X = x ) = 1IN, x = 1 . . . . . N, and points out that

~(X)=

2X-

2No

1

for X q : N o , X --/=No + 1 for X = No, No + 1

is a UMVUE of N while X is not a complete sufficient statistic for Po (or N). Rohatgi [1976, 356-358] reproduces this result and states that it is an example of the phenomenon considered in this note. However, it is not. In fact, as is not hard to verify, T (X) is a complete sufficient statistic for Po. In view of the results in this note, it is natural to ask whether a function of a UMVUE is always a UMVUE. The answer is negative and was given by the help of an example already in Bahadur [1957]. It is appropriate to end the note by just mentioning that important generalizations of Bahadur's results have been made by Schmetterer, Strasser, Torgersen, and others. Many references can be found in Kozek [1980]. Appendix Let P be a family of probability measures P. The result that a boundedly complete (P-)sufficient o-field A is necessarily minimal sufficient (i.e., if AI is also sufficient, then A c A1 a.s. P) is a known result but difficult to find in the standard statistical literature. In e.g. Schmetterer [1974, p. 220] as in Lehmann/Scheff~ [1950] it is given with the superfluous additional assumption that a minimal sufficient o-field exists and also Zacks [1971, p. 73] makes this assumption implicitly. The following proof, based on the Lehmann-Scheff~ theorem and found in conversations with A. Kozek, is so simple that the present author cannot help presenting it. Proof: Let A E A be arbitrary. Then, by the Lehmann-Scheff~ theorem, the indicator function 1A is a UMVUE. But obviously, by the Rao-Blackwell argument, E [IA I Ax ] is also an unbiased estimator o f E e [1A ] with variance less than or equal to that of

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I A . Hence also E [IA I A1 ] is a UMVUE, and by the well-known uniqueness of a

UMVUE we have 1A = E [IA [ AI ] a.s.P. In conclusion, A c A1 a.s. P, i.e., A is minimal sufficient.

Acknowledgement I want to thank A. Kozek, B. Olofsson, E. Torgersen, and the referee for their assistance. Remarks Added in Proof

a) The well-known result that X is a UMVUE of the mean of a completely unknown distribution provides perhaps the simplest example of the studied phenomenon; b) Another proof of the result in the Appendix can be found in Heyer, H. : Mathematische Theorie statistischer Experimente. Berlin-Heidelberg-New York 1973, 25-26.

References Bahadur, R.R. : On unbiased estimates of uniformly minimum variance. Sankhy~ 18, 1957, 211 224.

Barndorff-Nielsen, O. : Information and Exponential Families in Statistical Theory. Chichester 1978.

Epstein, B., and M. Sobel: Some theorems relevant to life testing from an exponential distribution. Ann. Math. Star. 25, 1954, 373-381.

Fraser, D.A.S. : Sufficient statistics with nuisance parameters. Ann. Math. Stat. 27, 1956, 838-842. Kozek, A. : On two necessary o-fields and on universal loss functions. Prob. Math. Star. 1, 1980, 29-47.

Lehmann, E.L., and H. Scheff~: Completeness, similar regions and unbiased estimation, Part I. Sankhy~ 10, 1950, 305-340.

Rao, C.R. : Linear Statistical Inference and Its Applications. New York, 2rid ed., 1973. Rohatgi, V.K. : An Introduction to Probability Theory and Mathematical Statistics. New York 1976. Schmetterer, L. : Introduction to Mathematical Statistics. Berlin-Heidelberg-New York 1974. Stigler, S.M.: Completeness and unbiased estimation. Amer. Stat. 26, 1972, 28-29. Zacks, S. : The Theory of Statistical Inference. New York 1971. Received, July 7, 1980 (revised version July 30, 1980)