Summary
Fix a family C of continuous distributions on the line. Sufficient and (different) necessary conditions on C are given in order that the sample distribution function be an optimal estimator in the asymptotic minimax sense. The abstract results are illustrated by a variety of concrete families C that have arisen in the literature; some of these illustrations settle known, but previously unsolved, problems. Methods involve systematic consideration of statistical experiments whose parameter lies in a Hilbert space, and the theory of abstract Wiener spaces.
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Barlow, R.E., Bartholomew, D.J., Bremmer, J.M., Brunk, H.D.: Statistical inference under order restrictions. New York: Wiley 1972
Beran, R.J.: Estimating a distribution function. Ann. Statist. 5, 400–404 (1977)
Beran, R.J.: Rank spectral processes and tests for serial dependence. Ann. Math. Statist. 43, 1749–1766 (1972)
Chernoff, H.: Large sample theory: parametric case. Ann. Math. Statist. 27, 1–22 (1956)
Dudley, R.M., Feldman, J., LeCam, L.: On semi-norms and probabilities and abstract Wiener spaces. Ann. Math. 93, 390–408 (1971)
Dvoretsky, A., Kiefer, J., Wolfowitz, J.: Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator. Ann. Math. Statist. 27, 642–669 (1956)
Feldman, J.: Equivalence and perpendicularity of Gaussian processes. Pacific J. Math. 8, 699–708 (1958)
Gelfand, I.M., Vilenkin, N.Ya.: Generalized Functions, Vol. 4. New York: Academic Press 1964
Gross, L.: Abstract Wiener spaces, Proc. Fifth Berkeley Sympos. Math. Statist. Probab. Univ. Calif. 2, 31–42 (1965)
Hajek, J.: A property of J-divergences of marginal probability distributions. Czechoslovak Math. J. 8, 460–63 (1958)
Hajek, J.: Local asymptotic minimax and admissibility in estimation. Proc. sixth Berkeley Sympos. Math. Statist. Probab. Univ. Calif. 1, 175–194 (1972)
Hajek, J., Sidak, Z.: Theory of Rank Tests. New York: Academic Press 1967
Kallianpur, G.: Abstract Wiener processes and their reproducing kernel Hilbert spaces. Z. Wahrscheinlichkeitstheorie verw. Gebiete 17, 113–123 (1971)
Kiefer, J., Wolfowitz, J.: Asymptotically minimax estimation of concave and convex distribution functions. Z. Wahrscheinlichkeitstheorie verw. Gebiete 34, 73–85 (1976)
Kuo, Hui-Hsiung: Gaussian Measures in Banach Spaces. New York: Springer 1975
LeCam, L.: Limits of experiments. Proc. Sixth Berkeley Sympos. Math. Statist. Probab. Univ. Calif. 1, 245–61 (1972)
LeCam, L.: Asymptotic Decision Theory. Preprint
LeCam, L.: On some results of J. Hajek concerning asymptotic normality. Preprint
LeCam, L.: Sufficiency and asymptotic sufficiency. Ann. Math. Statist. 35, 1419–1455 (1964)
LeCam, L.: Théorie Asymptotique de la Decision Statistique Les Presses de l'Universite de Montreal. Montreal, 1968
LeCam, L.: Convergence of estimates under dimensionality restrictions. Ann. Statist. 1, 38–53 (1973)
Moussatat, M.: On the asymptotic theory of statistical experiments and some of its applications. Thesis, Univ. of Calif., Berkeley, 1976
Rosenblatt, M.: Curve estimates. Ann. Math. Statist. 42, 1815–42 (1971)
Skorokhod, A.V.: Integration in Hilbert Space. K. Wickwire transl. New York: Springer 1974
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Millar, P.W. Asymptotic minimax theorems for the sample distribution function. Z. Wahrscheinlichkeitstheorie verw Gebiete 48, 233–252 (1979). https://doi.org/10.1007/BF00537522
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DOI: https://doi.org/10.1007/BF00537522