Estimation variances for estimators of product densities and pair correlation functions of planar point processes

Abstract

Approximations of the estimation variances of kernel estimators of the pair correlation function and the product density of a planar Poisson process are given. Furthermore, a heuristic approximation of the estimation variance of an estimator of the pair correlation function of a "general" planar point process is suggested. All formulae have been tested by simulation experiments. Second order characteristics play an important role in point process statistics. Usually, Ripley's K-function and the L-function (see Ripley (1981) and Stoyan st al. (1987)) are used for goodness-of-fit tests and parameter estimation, while the product density 0(t) and the pair correlation function g(t) are used in exploratory data analysis. The form of these functions helps to understand the type of interac- tion in the point pattern and to find suitable point process models. In particular, minima and maxima (if existing) of the pair correlation function may give valuable information on the strength of order. Since estimated second order characteristics deviate from their theoretical counterparts because of statistical fluctuations, error bounds for these functions are very important. For example, they are needed to distinguish between statisti- cal fluctuations in an estimated pair correlation function and peaks which are due to real properties of the point process under study. Until now, variances of esti- mation for second order characteristics are known only in particular cases. Ripley (1988) has given such variances for a series of estimators of the K-function for the Poisson process. This paper gives estimation variances for product densities and pair correla- tion functions. First, in analogy with to Ripley's (1988) calculations, estimation variances in the Poisson process case are derived. The formulae obtained are quite