On the expected value and variance for an estimator of the spatio-temporal product density function

Second-order characteristics are used to analyse the spatio-temporal structure of the under- lying point process, and thus these methods provide a natural starting point for the analysis of spatio-temporal point process data. We restrict our attention to the spatio-temporal product density function, and develop a non-parametric edge-corrected kernel estimate of the product density under the second-order intensity-reweighted stationary hypothesis. The expectation and variance of the estimator are obtained, and closed form expressions derived under the Poisson case. A detailed simulation study is presented to compare our close expression for the variance with estimated ones for Poisson cases. The simulation experiments show that the theoretical form for the variance gives acceptable values, which can be used in practice. Finally, we apply the resulting estimator to data on the spatio-temporal distribution of invasive meningococcal disease in Germany.Francisco J. Rodríguez-Cortés’s research was supported by grant P1-1B2012-52. Mohammad Ghorbani’s research was supported by the Center for Stochastic Geometry and Advanced Bioimaging, funded by a grant from the Villum Foundation. Jorge Mateu’s research was supported by grant MTM2010-14961 from Ministery of Education