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2010

Challenging the empirical mean and empirical variance: a deviation study

Catoni, Olivier

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We present new M-estimators of the mean and variance of real valued random variables, based on PAC-Bayes bounds.

We analyze the non-asymptotic minimax properties of the deviations of those estimators for sample distributions having either a bounded variance or a bounded variance and a bounded kurtosis.
Under those weak hypotheses, allowing for heavy-tailed distributions, we show that the worst case deviations of the empirical mean are suboptimal.
We prove indeed that for any confidence level, there is some M-estimator whose deviations are of the same order as the deviations of the empirical mean of a Gaussian statistical sample, even when the statistical sample is instead heavy-tailed.

Challenging the empirical mean and empirical variance: a deviation study · Around

Built on

Robust estimation of a location parameter
Peter J. Huber · 1964
Earlier work this paper cites.
Adaptive maximum likelihood estimators of a location parameter
Charles J. Stone · 1975
Earlier work this paper cites.
An efficient and robust adaptive estimator of location
Rudolf Beran · 1978
Earlier work this paper cites.
Robust Statistics
Peter J. Huber · 1981
Earlier work this paper cites.
On adaptive estimation
P. J. Bickel · 1982
Earlier work this paper cites.

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Then

Statistical Learning Theory and Stochastic Optimization, Lectures on Probability Theory and Statistics, École d’Été de Probabilités de Saint-Flour XXXI – 2001
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Robust linear least squares regression
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