Nonparametric inference for Markovian interval processes
A multivariate counting process \(N=(N_1,\ldots ,N_p)\) is considered. It is assumed that each individual intensity depends (only) on the current time and on the time elapsed from the last counts of other components (such a process is called a Markov interval process). The times passed since preceding counts, \(z(t)=(z_1(t),\ldots ,z_p(t))\), are treated as covariates in a nonparametric hazard regression model \(\lambda _i(t)=\alpha _i(t,z(t)),\;i=1,\ldots ,p\). The procedure of estimation constructs first the nonparametric estimator of a double cumulative hazard rate [cf. the references to recent papers of the author and \textit{I.W. McKeague}, e.g., Scand. J. Stat. 18, No. 3, 177-195 (1991; Zbl 0803.62038)], then estimators of functions \(\alpha _i\) are obtained via smoothing. Two main theorems prove the large sample properties, namely the consistency of estimates (uniform on compact sets), and asymptotic normality of estimated ``increments'' (w.r. to variable \(z\)) of double cumulative intensities. The approach is then utilized for the construction of tests of hypotheses, namely for tests of a) linearity of dependence on \(z\), b) independence of components. Finally, the method is applied to modelling the relationship between neurons organized to a net where activation of each cell depends on impulses from the others. Up to now the description of such a network has mostly been based on cross-correlation analysis.
scientific article; zbMATH DE number 1134818