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A Catoni mean estimator estimates an expected value from independent and identically distributed random variables with finite variance, using a nonlinear score instead of directly averaging the observations. Choose a continuous monotone function $\psi$ satisfyingand solve ${\sum }_i\psi (\alpha (X_i-\hat{\mu }))=0$ for $\alpha >0$.

For known variance $v>0$, set $L=\log (1/\epsilon )$, $0<\epsilon <1/2$, and assume $n>2L$. The choicesgive $\mathbb{P}(|\hat{\mu }-\mu |\le \eta )\ge 1-2\epsilon$. The quadratic envelope controls exponential moments of the transformed score using only second moments; Markov's inequality and score monotonicity then bracket its root. The tuning depends on the variance and confidence level in this version.^[ref]

The estimating score $\psi$ is not automatically a normalized population influence function. This procedure targets the mean of the sampled probability distribution; it does not identify an uncontaminated component's location without additional assumptions.

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]