Kernel density estimation for stationary random fields

Mohamed El Machkouri (1)

1 LMRS - Laboratoire de Mathématiques Raphaël Salem

In this paper, under natural and easily verifiable conditions, we prove the $\mathbb{L}^1$-convergence and the asymptotic normality of the Parzen-Rosenblatt density estimator for stationary random fields of the form $X_k = g\left(\varepsilon_{k-s}, s \in \Z^d \right)$, $k\in\Z^d$, where $(\varepsilon_i)_{i\in\Z^d}$ are i.i.d real random variables and $g$ is a measurable function defined on $\R^{\Z^d}$. Such kind of processes provides a general framework for stationary ergodic random fields. A Berry-Esseen's type central limit theorem is also given for the considered estimator.

Mots clés: Central limit theorem; spatial processes; m-dependent random fields; physical dependence measure; nonparametric estimation; kernel density estimator; rate of convergence

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Mohamed El Machkouri. Kernel density estimation for stationary random fields. 2014. ⟨hal-00622861v5⟩