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

Abstract

Approximation, estimation variance, pair correlation function, Poisson process, product density, simulation,

Modelling, Estimation and Applications of Second-Order Spatio-Temporal Characteristics of Point Processes

“…However, our estimator (2.14) does not have the same structure as the quantity T in Ripley (1989), Stoyan et al (1993a) and Daley and Vere-Jones (2003)), because w(u, v) = w(v, u), i.e. the edge-correction function is not symmetric implying that the term within the sum in (2.14) is not symmetric.…”

“…Then Stoyan et al (1993b) approximated the variance of spatial product densities, and Cressie and Collins (2001a,b) obtained close expressions for the expected and variance values of the local spatial product densities. Nothing has been developed in the spatio-temporal context.…”

“…For spatial case, using Epanechnikov kernel Stoyan et al (1993b) have shown that For temporal case and using uniform kernel, by the same methods as in Stoyan et al (1993b), it is easy to show that…”

“…To obtain an estimator for the variance of the second-order product density function Stoyan et al (1993a) and Stoyan and Stoyan (2000) proposed different methodologies using the set covariance function as edgecorrection. However, our estimator (2.14) does not have the same structure as the quantity T in Ripley (1989), Stoyan et al (1993a) and Daley and Vere-Jones (2003)), because w(u, v) = w(v, u), i.e.…”

“…First-and second-order spatio-temporal separability are introduced. We extend the original ideas of Stoyan et al (1993b) to the spatio-temporal case to develop exact and close expressions of the expectation and variance of the proposed estimator. Note that since estimated second-order characteristics deviate from their theoretical counterparts because of statistical fluctuations, error bounds for these functions are important.…”