On Some Statistical Properties of the Spatio-Temporal Product Density

We present an extension of the non-parametric edge-corrected Ohser-typekernel estimator for the spatio-temporal product density function. We derivethe mean and variance of the estimator and give a closed-form approximationfor a spatio-temporal Poisson point process. Asymptotic properties ofthis second-order characteristic are derived, using an approach based onmartingale theory. Taking advantage of the convergence to normality,confidence surfaces under the homogeneous Poisson process are built. Asimulation study is presented to compare our approximation for the variancewith Monte Carlo estimated values. Finally, we apply the resulting estimatorand its properties to analyse the spatio-temporal distribution of the invasivemeningococcal disease in the Rhineland Regional Council in Germany.