Limit Distribution Theory for Maximum Likelihood Estimation of a Log-Concave Density.

نویسندگان

  • Fadoua Balabdaoui
  • Kaspar Rufibach
  • Jon A Wellner
چکیده

We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density, i.e. a density of the form f(0) = exp varphi(0) where varphi(0) is a concave function on R. Existence, form, characterizations and uniform rates of convergence of the MLE are given by Rufibach (2006) and Dümbgen and Rufibach (2007). The characterization of the log-concave MLE in terms of distribution functions is the same (up to sign) as the characterization of the least squares estimator of a convex density on [0, infinity) as studied by Groeneboom, Jongbloed and Wellner (2001b). We use this connection to show that the limiting distributions of the MLE and its derivative are, under comparable smoothness assumptions, the same (up to sign) as in the convex density estimation problem. In particular, changing the smoothness assumptions of Groeneboom, Jongbloed and Wellner (2001b) slightly by allowing some higher derivatives to vanish at the point of interest, we find that the pointwise limiting distributions depend on the second and third derivatives at 0 of H(k), the "lower invelope" of an integrated Brownian motion process minus a drift term depending on the number of vanishing derivatives of varphi(0) = log f(0) at the point of interest. We also establish the limiting distribution of the resulting estimator of the mode M(f(0)) and establish a new local asymptotic minimax lower bound which shows the optimality of our mode estimator in terms of both rate of convergence and dependence of constants on population values.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Limit Distribution Theory for Maximum Likelihood Estimation of a Log-concave Density by Fadoua

We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density, that is, a density of the form f0 = expφ0 where φ0 is a concave function on R. The pointwise limiting distributions depend on the second and third derivatives at 0 of Hk , the “lower invelope” of an integrated Brownian motion process minus a drift term depending on the number of vani...

متن کامل

Maximum likelihood estimation of a log-concave density and its distribution function: Basic properties and uniform consistency

We study nonparametric maximum likelihood estimation of a log–concave probability density and its distribution and hazard function. Some general properties of these estimators are derived from two characterizations. It is shown that the rate of convergence with respect to supremum norm on a compact interval for the density and hazard rate estimator is at least (log(n)/n) and typically (log(n)/n...

متن کامل

APPROXIMATION AND ESTIMATION OF s-CONCAVE DENSITIES VIA RÉNYI DIVERGENCES.

In this paper, we study the approximation and estimation of s-concave densities via Rényi divergence. We first show that the approximation of a probability measure Q by an s-concave density exists and is unique via the procedure of minimizing a divergence functional proposed by [Ann. Statist.38 (2010) 2998-3027] if and only if Q admits full-dimensional support and a first moment. We also show c...

متن کامل

logcondens: Computations Related to Univariate Log-Concave Density Estimation

Maximum likelihood estimation of a log-concave density has attracted considerable attention over the last few years. Several algorithms have been proposed to estimate such a density. Two of those algorithms, an iterative convex minorant and an active set algorithm, are implemented in the R package logcondens. While these algorithms are discussed elsewhere, we describe in this paper the use of t...

متن کامل

Computing confidence intervals for log-concave densities

Log-concave density estimation is considered as an alternative for kernel density estimations which does not depend on tuning parameters. Pointwise asymptotic theory has been already developed for the nonparametric maximum likelihood estimator of a log-concave density. Here, the practical aspects of this theory are studied. In order to obtain a confidence interval, estimators of the constants a...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:
  • Annals of statistics

دوره 37 3  شماره 

صفحات  -

تاریخ انتشار 2009