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Vitesse de convergence en M-estimation de données markoviennes

Abstract : Let $\{X_n\}_{n\ge 0}$ be a $V$-geometrically ergodic Markov chain with $V\geq 1$ some fixed unbounded real-valued function and consider $M_n(\alpha) = n^{-1} \sum_{k=1}^n F(\alpha,X_{k-1},X_k)$, $\alpha\in\mathcalA\in \mathbbR$ for some real-valued functional $F(\cdot,\cdot,\cdot)$. Define the $M-$estimator $\widehat \alpha_n $ such that $M_n(\widehat \alpha_n) \leq \min_{\alpha\in\mathcalA} M_n(\alpha) + c_n$ with $c_n$, $n\geq 1$ some sequence of real numbers decreasing to zero. Under some standard regularity assumptions, close to that of the i.i.d case, and under the moment assumption $$\left(\bigg|\frac{\partial F}{\partial\alpha}(\alpha,x,y)\bigg| + \bigg|\frac{\partial^2 F}{\partial\alpha^2}(\alpha,x,y)\bigg|\right)^{3+\varepsilon} \leq C\, (V(x) + V(y))$$ for some constants $\varepsilon>0$ and $C>0$, the estimator $\widehat\alpha_n $ satisfies a Berry-Esseen theorem uniformly with respect to the underlying probability distribution of the Markov chain.
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Loïc Hervé, James Ledoux, Valentin Patilea. Vitesse de convergence en M-estimation de données markoviennes. 42èmes Journées de Statistique, 2010, Marseille, France, France. ⟨inria-00494678⟩

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