Measuring the association of a time series and a point process

Abstract

We consider a bivariate stochastic process where one component is an ordinary time series and the other is a point process. In the stationary case, a useful measure of the association of the time series and the point process is provided by a conditional intensity function, ṙ11(x;u), which gives the intensity with which events occur near time t given that the time series takes on a value x at time t + u. In this paper we consider the estimation of the function ṙ11(x;u) and certain related functions that are also useful in partially characterizing the degree of interdependence of the time series and the point process. Histogram and smoothed histogram-type estimates are proposed and asymptotic distributions of these estimates are derived. We also discuss an application of the estimation theory to the analysis of some data from a neurophysiological study.