Summary
In the present paper the problem of point estimation is considered in terms of risk functions, without the customary restriction to unbiased estimates. It is shown that, whenever the loss is a convex function of the estimate, it suffices from the risk viewpoint to consider only nonrandomized estimates. For a number of specific problems the minimax estimates are found explicitly, using the squared error as loss. Certain minimax prediction problems are also solved.
This work was supported in part by the Office of Naval Research.
Actually, the principle of minimum variance unbiased estimation goes back to Gauss. For discussions of the history of these ideas, see E. CzuberS, Theorie der Beobachtungsfehler, Leipzig, 1891, and R. L. PLACKETT, “A historical note on the method of least squares”, Biometrika, Vol. 36 (1950), p. 458.
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References
G. W. Brown, “On small 6ample estimation,” Annals of Math. Stat., Vol. 18 (1949), p. 514.
A. Wald, “Contributions to the theory of statistical estimation and testing hypotheses,” Annals of Math. Stat., Vol. 10 (1939), p. 299.
A. Wald, On the Principles of Statistical InferenceNotre Dame Math. Lectures, No. 1 (1942).
A. Wald, “Statistical decision functions which minimize the maximum risk,” Annals of Math., Vol. 46 (1945), p. 265.
A. Wald, “Statistical decision functions” Annals of Math. Stat., Vol.-20 (1949), p. 165.
C. Stein and A. Wald, “Sequential confidence intervals for the mean of a normal dis tribution with known variance,” Annals of Math. Stat., Vol. 18 (1947), p. 427.
P. R. Halmos, “The theory of unbiased estimation,” Annals of Math. Stat., Vol. 17 (1946), p. 34.
E. L. Lehmann and C. Stein, “Most powerful tests of composite hypotheses. I. Normal distributions,” Annals of Math.’Stat., Vol. 19 (1948), p. 495.
A. Wald, “An essentially complete class of admissible decision functions,” Annals of Math. Stat., Vol. 18 (1947j, p. 549.
A. Kolmogoroff, Grundbegriffe der Wahrscheinlichkeitsrechnung Berlin, 1933.
D. Blackwell, “Conditional expectation and unbiased sequential estimation,” Annals of Math. Stat., Vol. 18 (1947), p. 105.
E. W. Barankin, “Extension of a theorem of Blackwell,” Annals of Math. Stat., Vol. 21 (1950), p. 280.
A. M. Mood, “On the dependence of sampling inspection plans upon population distributions,” Annals of Math. Stat., Vol. 14 (1943), p. 145.
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Hodges, J.L., Lehmann, E.L. (2012). Some Problems in Minimax Point Estimation. In: Rojo, J. (eds) Selected Works of E. L. Lehmann. Selected Works in Probability and Statistics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-1412-4_3
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