Details

Autor(en) / Beteiligte

• Shashiashvili, Malkhaz

Titel

Nonasymptotic bounds for the quadratic risk of the Grenander estimator

Ist Teil von

• Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, 2021-04, Vol.151 (2), p.464-472

Ort / Verlag

Edinburgh, UK: Royal Society of Edinburgh Scotland Foundation

Erscheinungsjahr

2021

Link zum Volltext

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• 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.