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Nonasymptotic bounds for the quadratic risk of the Grenander estimator

Published online by Cambridge University Press:  07 April 2020

Malkhaz Shashiashvili
Malkhaz Shashiashvili*
Affiliation:
Department of Mathematics, Faculty of Exact and Natural Sciences, Iv. Javakhishvili Tbilisi State University, 3 University Str., 0143Tbilisi, Georgia (malkhaz.shashiashvili@tsu.ge)
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Abstract

There is an enormous literature on the so-called Grenander estimator, which is merely the nonparametric maximum likelihood estimator of a nonincreasing probability density on [0, 1] (see, for instance, Grenander (1981)), but unfortunately, there is no nonasymptotic (i.e. for arbitrary finite sample size n) explicit upper bound for the quadratic risk of the Grenander estimator readily applicable in practice by statisticians. In this paper, we establish, for the first time, a simple explicit upper bound 2n−1/2 for the latter quadratic risk. It turns out to be a straightforward consequence of an inequality valid with probability one and bounding from above the integrated squared error of the Grenander estimator by the Kolmogorov–Smirnov statistic.

Keywords

Grenander estimatornonasymptotic boundsnonparametric estimationquadratic risk boundsKolmogorov–Smirnov statistic
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 2 , April 2021 , pp. 464 - 472
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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Footnotes

This paper is dedicated to Professor Estate Khmaladze on the occasion of his 75th birthday.

References

1Birgé, L.. The Grenander estimator: a nonasymptotic approach. Ann. Statist. 17 (1989), 1532–1549.CrossRefGoogle Scholar
2Birgé, L.. Estimation of unimodal densities without smoothness assumptions. Ann. Statist. 25 (1997), 970–981.CrossRefGoogle Scholar
3Devroye, L. and Lugosi, G.. Combinatorial methods in density estimation. Springer Series in Statistics (New York: Springer-Verlag, 2001).CrossRefGoogle Scholar
4Grenander, U.. On the theory of mortality measurement. Part II. Skand. Aktuarietidskr. 1956 (1957), 125–153.Google Scholar
5Grenander, U.. Abstract inference. Wiley Series in Probability and Mathematical Statistics (New York: John Wiley & Sons, Inc., 1981).Google Scholar
6Groeneboom, P., Estimating a monotone density. In Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer, vol. II (Berkeley, Calif., 1983). Wadsworth Statist./Probab. Ser., pp. 539–555 (Belmont, CA: Wadsworth, 1985).Google Scholar
7Groeneboom, P. and Pyke, R.. Asymptotic normality of statistics based on the convex minorants of empirical distribution functions. Ann. Probab. 11 (1983), 328–345.CrossRefGoogle Scholar
8Hewitt, E. and Stromberg, K.. Real and abstract analysis. A modern treatment of the theory of functions of a real variable. Graduate Texts in Mathematics, vol. 25 (New York, Heidelberg: Springer-Verlag, 1975).Google Scholar
9Kiefer, J. and Wolfowitz, J.. Asymptotically minimax estimation of concave and convex distribution functions. Z. Wahrscheinlichkeitstheorie Verw. Gebiete 34 (1976), 73–85.CrossRefGoogle Scholar
10Prakasa Rao, B. L. S.. Estimation of a unimodal density. Sankhyā Ser. A 31 (1969), 23–36.Google Scholar
11van der Vaart, A. W.. Estimation of a unimodal density. Cambridge Series in Statistical and Probabilistic Mathematics, vol. 3 (Cambridge: Cambridge University Press, 1998).Google Scholar