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• W2963518408 abstract "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 $widehat{F}_{n}$ of a naive estimator $F_{n}$ of the integrated curve $F$ corresponding to $f$. We prove that the supremum distance between $widehat{F}_{n}$ and $F_{n}$ is of the order $O_{p}(n^{-1}logn)^{2/(4-tau)}$, for some $tauin[0,4)$ that characterizes the tail probabilities of an approximating process for $F_{n}$. In typical examples, the approximating process is Gaussian and $tau=1$, in which case the convergence rate $n^{-2/3}(log n)^{2/3}$ 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 $F_{n}$, in which case $tau=2$, leading to a faster rate $n^{-1}log n$, 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." @default.
• W2963518408 created "2019-07-30" @default.
• W2963518408 creator A5061249254 @default.
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• W2963518408 date "2014-01-01" @default.
• W2963518408 modified "2023-10-12" @default.
• W2963518408 title "A Kiefer-Wolfowitz type of result in a general setting, with an application to smooth monotone estimation" @default.
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• W2963518408 doi "https://doi.org/10.1214/14-ejs958" @default.
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