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Minimum variance and the estimation of several parameters

Published online by Cambridge University Press:  24 October 2008

C. Radhakrishna Rao
C. Radhakrishna Rao
Affiliation:
King's CollegeCambridge
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Extract

With the help of certain inequalities concerning the elements of the dispersion matrix of a set of statistics, and of the information matrix, the following results have been proved. Some of these inequalities are extensions of results given by Fisher (1) in the case of a single parameter.

(i) Efficient statistics are explicit functions of the minimal set of sufficient statistics.

(ii) Functions of the minimal set of sufficient statistics, satisfying the property of uniqueness defined in the text, are best unbiased estimates. Under certain conditions estimates possessing exactly the minimum possible variance can be obtained by the method of maximum likelihood.

(iii) In large samples maximum likelihood estimates supply efficient statistics in the case of several parameters.

(iv) The importance of replacing the sample by an exhaustive set of sufficient statistics (referred to in this paper as the minimal set) as a first step in any methodological problem has been stressed by R. A. Fisher in various articles and lectures. The above discussion supplies a formal demonstration of this view so far as the problem of estimation is concerned.

Type
Research Notes
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 43 , Issue 2 , April 1947 , pp. 280 - 283
Copyright
Copyright © Cambridge Philosophical Society 1947

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References

REFERENCES

(1)Fisher, R. A.The logic of inductive inference. J. Roy. Static. Soc. 98 (1935), 39–82.CrossRefGoogle Scholar
(2)Geary, R. C.The estimation of many parameters. J. Roy. Static. Soc. 105 (1942), 213–17.CrossRefGoogle Scholar
(3)Rao, C. R.Information and the accuracy attainable in the estimation of several parameters. Calcutta Math. Bull. 37 (1945), 81–91.Google Scholar
(4)Wald, A.Tests of statistical hypotheses concerning several parameters when the number of observations is large. Trans. American Math. Soc. 54 (1943), 426–82.CrossRefGoogle Scholar