Asymptotic normality of multivariate frequency polygons for stationary random fields

Abstract

The purpose of this paper is to investigate the asymptotic normality of the multivariate frequency polygon as a density estimator of a stationary mixing random field indexed by multidimensional lattice points space \(\mathbb {Z}^N\). Results on weak convergence of the estimator are established, including a simple analytic form for its asymptotic variance. A consistent estimator is proposed for this variance. Simulations confirm the theoretical results. Bias correction and appropriate choices of the bandwidths are discussed. The results apply to many spatial random models, such as spatial autoregressive models, spatio-temporal geostatistical models, spatial epidemiology.