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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 ([email protected])
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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.

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