= Catoni mean estimator
{c}
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> $\psi$ satisfying
$$
-\log(1-u+u^2/2)\leq\psi(u)\leq\log(1+u+u^2/2),
$$
and solve $\sum_i\psi(\alpha(X_i-\widehat\mu))=0$ for $\alpha>0$.
For known <variance> $v>0$, set $L=\log(1/\varepsilon)$, $0<\varepsilon<1/2$, and assume $n>2L$. The choices
$$
\eta=\sqrt{\frac{2vL}{n(1-2L/n)}},\qquad \alpha=\sqrt{\frac{2L}{n(v+\eta^2)}}
$$
give $\mathbb P(|\widehat\mu-\mu|\leq\eta)\geq1-2\varepsilon$. 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.<Challenging the empirical mean and empirical variance: a deviation study by Olivier Catoni>[Section 2, equations 2.1–2.3 and Proposition 2.4]{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.
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