Density estimation for spatial linear processes

Abstract

The problem of estimating the marginal densities of a spatial linear process, observed over a grid of $\mathbb {Z}^N$, is considered. Under general conditions, kernel density estimators computed at any $k$-tuple of sites are shown to be asymptotically multivariate normal. Their limiting covariance matrix is also computed. Despite the huge development of nonparametric estimation methods in the analysis of time series data, little has so far been done to introduce them into the context of random fields. The generalization indeed is far from trivial since the points of $\mathbb {Z}^N$ do not have a natural ordering when $N>1$. No mixing conditions are required, but linearity is assumed.