On the optimal estimation of probability measures in weak and strong topologies

Research output: Contribution to journalArticle

9 Citations (Scopus)

Abstract

Given random samples drawn i.i.d. from a probability measure P (defined on say, Rd ), it is well-known that the empirical estimator is an optimal estimator of 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 R that are optimal in strong topology are also optimal in || ||F for certain choices of F such that || ||F metrizes the weak topology, where P|| f || := sup{ f dP: f € F}. In this paper, we investigate this problem of optimal estimation in weak and strong topologies by choosing F to be a unit ball in a reproducing kernel Hilbert space (say F H defined over Rd ), where this choice is both of theoretical and computational interest. Under some mild conditions on the reproducing kernel, we show that || ||FH metrizes the weak topology and the kernel density estimator (with L1 optimal bandwidth) estimates P at dimension independent optimal rate of n -1/2 in || ||FH along with providing a uniform central limit theorem for the kernel density estimator.

Original languageEnglish (US)
Pages (from-to)1839-1893
Number of pages55
JournalBernoulli
Volume22
Issue number3
DOIs
StatePublished - Aug 1 2016

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Optimal Estimation
Probability Measure
Density Estimator
Weak Topology
Topology
Kernel Density Estimator
Empirical Estimator
Wavelet Estimator
Minimax Rate
Total Variation Distance
Estimator
Optimal Bandwidth
Reproducing Kernel Hilbert Space
Optimal Rates
Reproducing Kernel
Consistent Estimator
Unit ball
Central limit theorem
Ball
kernel

All Science Journal Classification (ASJC) codes

  • Statistics and Probability

Cite this

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abstract = "Given random samples drawn i.i.d. from a probability measure P (defined on say, Rd ), it is well-known that the empirical estimator is an optimal estimator of 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 Gin{\'e} and Nickl that these density estimators on R that are optimal in strong topology are also optimal in || ||F for certain choices of F such that || ||F metrizes the weak topology, where P|| f || := sup{ f dP: f € F}. In this paper, we investigate this problem of optimal estimation in weak and strong topologies by choosing F to be a unit ball in a reproducing kernel Hilbert space (say F H defined over Rd ), where this choice is both of theoretical and computational interest. Under some mild conditions on the reproducing kernel, we show that || ||FH metrizes the weak topology and the kernel density estimator (with L1 optimal bandwidth) estimates P at dimension independent optimal rate of n -1/2 in || ||FH along with providing a uniform central limit theorem for the kernel density estimator.",
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On the optimal estimation of probability measures in weak and strong topologies. / Sriperumbudur, Bharath Kumar.

In: Bernoulli, Vol. 22, No. 3, 01.08.2016, p. 1839-1893.

Research output: Contribution to journalArticle

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