A Kiefer-Wolfowitz type of result in a general setting, with an application to smooth monotone estimation

• Published in: Electronic journal of statistics Vol. 8; no. 2
• Main Authors: Durot, Cécile, Lopuhaä, Hendrik P.
• Format: Journal Article
• Language: English
• Published: Shaker Heights, OH : Institute of Mathematical Statistics 01.01.2014
• Subjects: Statistics; Statistics Theory; isotonic regression; mono-tone density; monotone failure rate; monotone regression; least concave majorant
• ISSN: 1935-7524; 1935-7524
• DOI: 10.1214/14-EJS958

We consider Grenander type estimators for monotone functions f in a very general setting, which includes estimation of monotone regression curves, monotone densities, and monotone failure rates. These estimators are defined as the left-hand slope of the least concave majorant Fn of a naive estimator Fn of the integrated curve F corresponding to f. We prove that the supremum distance between Fn and Fn is of the order Op((log n)/n)^ {2/(4−τ)} , for some τ ∈ [0, 4) that characterizes the tail probabilities of an approximating process for Fn. In typical examples, the approximating process is Gaussian and τ = 1, in which case the convergence rate is ((log n)/n)^ 2/3 is in the same spirit as the one obtained by Kiefer and Wolfowitz [9] for the special case of estimating a decreasing density. We also obtain a similar result for the primitive of Fn, in which case τ = 2, leading to a faster rate (log n)/n, also found by Wang and Woodfroofe [22]. As an application in our general setup, we show that a smoothed Grenander type estimator and its derivative are asymptotically equivalent to the ordinary kernel estimator and its derivative in first order. MSC 2010 subject classifications: Primary 62G05; secondary 62G07 62G08 62N02.