Olivier Catoni develops methods for robust statistics. His work on the mean and variance constructs M-estimators with deviation guarantees for heavy-tailed distributions under moment assumptions.[ref]
The sample mean lets a single very large observation exert arbitrarily large influence. Catoni's construction replaces a linear estimating equation by a controlled nonlinear function of each residual. With an appropriate tuning parameter, a mean estimator can achieve Gaussian-order deviation bounds at a chosen confidence level under bounded variance, even when the observations are not normally distributed. The tuning and assumptions are part of the guarantee; the result does not assert that one fixed setting automatically works at every confidence level.[ref]
Challenging the empirical mean and empirical variance: a deviation study by Olivier Catoni. Equations2.1–2.3 and Proposition2.4
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