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

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 $\hat{F}_n$ of a naive estimator $F_n$ of the integrated curve $F$ corresponding to $f$. We prove that the supremum distance between $\hat{F}_n$ and $F_n$ is of the order $O_p(n^{-1}\log n)^{2/(4-\tau)}$, for some $\tau\in[0,4)$ 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 is $n^{-2/3}(\log n)^{2/3}$ is in the same spirit as the one obtained by Kiefer and Wolfowitz (1976) 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 (2007). 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

The behavior of the NPMLE of a decreasing density near the boundaries of the support

We investigate the behavior of the nonparametric maximum likelihood estimator $\hat{f}_n$ for a decreasing density $f$ near the boundaries of the support of $f$. We establish the limiting distribution of $\hat{f}_n(n^{-\alpha})$, where we need to distinguish between different values of $0<\alpha<1$. Similar results are obtained for the upper endpoint of the support, in the case it is finite. This yields consistent estimators for the values of $f$ at the boundaries of the support. The limit distribution of these estimators is established and their performance is compared with the penalized maximum likelihood estimator.Comment: Published at http://dx.doi.org/10.1214/009053606000000100 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Asymptotic normality of the LkL_k-error of the Grenander estimator

We investigate the limit behavior of the $L_k$-distance between a decreasing density $f$ and its nonparametric maximum likelihood estimator $\hat{f}_n$ for $k\geq1$. Due to the inconsistency of $\hat{f}_n$ at zero, the case $k=2.5$ turns out to be a kind of transition point. We extend asymptotic normality of the $L_1$-distance to the $L_k$-distance for $1\leq k<2.5$, and obtain the analogous limiting result for a modification of the $L_k$-distance for $k\geq2.5$. Since the $L_1$-distance is the area between $f$ and $\hat{f}_n$, which is also the area between the inverse $g$ of $f$ and the more tractable inverse $U_n$ of $\hat{f}_n$, the problem can be reduced immediately to deriving asymptotic normality of the $L_1$-distance between $U_n$ and $g$. Although we lose this easy correspondence for $k>1$, we show that the $L_k$-distance between $f$ and $\hat{f}_n$ is asymptotically equivalent to the $L_k$-distance between $U_n$ and $g$.Comment: Published at http://dx.doi.org/10.1214/009053605000000462 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

The limit distribution of the L∞L_{\infty}-error of Grenander-type estimators

Let $f$ be a nonincreasing function defined on $[0,1]$. Under standard regularity conditions, we derive the asymptotic distribution of the supremum norm of the difference between $f$ and its Grenander-type estimator on sub-intervals of $[0,1]$. The rate of convergence is found to be of order $(n/\log n)^{-1/3}$ and the limiting distribution to be Gumbel.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1015 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org