Nearest Neighbor Estimators for Random Fields

Generalizing the random sequence case, this study defines a k - NN density estimator for random variables with multidimensional lattice points serving as index values. The central result is that under random field stationary and mixing assumptions, as well as standard smoothness postulates, our k - NN estimate is found to be asymptotically normal. This result readily extends to NN-type estimates for jointly distributed random variables. For illustration, a simplified version of the k - NN estimator is applied to obtain the density estimate for a soil-moisture data set selected from the geostatistical literature.