scholarly journals On the Grenander estimator at zero

2011 ◽  
Vol 21 (2) ◽  
pp. 873 ◽  
Author(s):  
Fadoua Balabdaoui ◽  
Hanna Jankowski ◽  
Marios Pavlides ◽  
Arseni Seregin ◽  
Jon Wellner
Keyword(s):  
Grenander Estimator

scholarly journals Asymptotic normality of the $L_1$ error of the Grenander estimator

1999 ◽  
Vol 27 (4) ◽  
pp. 1316-1347 ◽  
Author(s):  
Piet Groeneboom ◽  
Gerard Hooghiemstra ◽  
Hendrik P. Lopuhaä
Keyword(s):  
Asymptotic Normality ◽  
Grenander Estimator

Asymptotic behavior of the grenander estimator at density flat regions

1999 ◽  
Vol 27 (3) ◽  
pp. 557-566 ◽  
Author(s):  
Chris Carolan ◽  
Richard Dykstra
Keyword(s):  
Asymptotic Behavior ◽  
Grenander Estimator

scholarly journals Asymptotic normality of the L k -error of the Grenander estimator

2005 ◽  
Vol 33 (5) ◽  
pp. 2228-2255 ◽  
Author(s):  
Vladimir N. Kulikov ◽  
Hendrik P. Lopuhaä
Keyword(s):  
Asymptotic Normality ◽  
Grenander Estimator

Nonasymptotic bounds for the quadratic risk of the Grenander estimator

2020 ◽  
pp. 1-9
Author(s):  
Malkhaz Shashiashvili
Keyword(s):  
Upper Bound ◽  
Finite Sample ◽  
Nonparametric Maximum Likelihood Estimator ◽  
Finite Sample Size ◽  
Kolmogorov Smirnov ◽  
Integrated Squared Error ◽  
Nonasymptotic Bounds ◽  
Grenander Estimator ◽  
First Time ◽  
Straightforward Consequence

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.

scholarly journals Inconsistency of bootstrap: The Grenander estimator

2010 ◽  
Vol 38 (4) ◽  
pp. 1953-1977 ◽  
Author(s):  
Bodhisattva Sen ◽  
Moulinath Banerjee ◽  
Michael Woodroofe
Keyword(s):  
Grenander Estimator
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