Open Access
August 2016 On the optimal estimation of probability measures in weak and strong topologies
Bharath Sriperumbudur
Bernoulli 22(3): 1839-1893 (August 2016). DOI: 10.3150/15-BEJ713


Given random samples drawn i.i.d. from a probability measure $\mathbb{P}$ (defined on say, $\mathbb{R}^{d}$), it is well-known that the empirical estimator is an optimal estimator of $\mathbb{P}$ in weak topology but not even a consistent estimator of its density (if it exists) in the strong topology (induced by the total variation distance). On the other hand, various popular density estimators such as kernel and wavelet density estimators are optimal in the strong topology in the sense of achieving the minimax rate over all estimators for a Sobolev ball of densities. Recently, it has been shown in a series of papers by Giné and Nickl that these density estimators on $\mathbb{R}$ that are optimal in strong topology are also optimal in $\Vert\cdot\Vert_{\mathcal{F} }$ for certain choices of $\mathcal{F}$ such that $\Vert\cdot\Vert_{\mathcal{F} }$ metrizes the weak topology, where $\Vert\mathbb{P} \Vert_{\mathcal{F} }:=\sup\{\int f\,\mathrm{d}\mathbb{P} \colon\ f\in\mathcal{F} \}$. In this paper, we investigate this problem of optimal estimation in weak and strong topologies by choosing $\mathcal{F}$ to be a unit ball in a reproducing kernel Hilbert space (say $\mathcal{F}_{H}$ defined over $\mathbb{R}^{d}$), where this choice is both of theoretical and computational interest. Under some mild conditions on the reproducing kernel, we show that $\Vert\cdot\Vert_{\mathcal{F}_{H}}$ metrizes the weak topology and the kernel density estimator (with $L^{1}$ optimal bandwidth) estimates $\mathbb{P}$ at dimension independent optimal rate of $n^{-1/2}$ in $\Vert\cdot\Vert_{\mathcal{F}_{H}}$ along with providing a uniform central limit theorem for the kernel density estimator.


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Bharath Sriperumbudur. "On the optimal estimation of probability measures in weak and strong topologies." Bernoulli 22 (3) 1839 - 1893, August 2016.


Received: 1 July 2014; Revised: 1 February 2015; Published: August 2016
First available in Project Euclid: 16 March 2016

zbMATH: 1360.62163
MathSciNet: MR3474835
Digital Object Identifier: 10.3150/15-BEJ713

Keywords: adaptive estimation , Bounded Lipschitz metric , Exponential inequality , kernel density estimator , Rademacher chaos , ‎reproducing kernel Hilbert ‎space , smoothed empirical processes , total variation distance , two-sample test , uniform central limit theorem , U-processes

Rights: Copyright © 2016 Bernoulli Society for Mathematical Statistics and Probability

Vol.22 • No. 3 • August 2016
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