Covariance density estimation for autoregressive spectral modelling of point processes

The use of autoregressive modelling has acquired great importance in time series analysis and in principle it may also be applicable in the spectral analysis of point processes with similar advantages over the nonparametric approach. Most of the methods used for autoregressive spectral analysis require positive semidefinite estimates for the covariance function, while current methods for the estimation of the covariance density function of a point process given a realization over the interval [0, T] do not guarantee a positive semidefinite estimate. This paper discusses methods for the estimation of the covariance density and conditional intensity function of point processes and present alternative computational efficient estimation algorithms leading always to positive semidefinite estimates, therefore adequate for autoregressive spectral analysis. Autoregressive spectral modelling of point processes from Yule-Walker type equations and Levinson recursion combined with the minimum AIC or CAT principle is illustrated with neurobiological data.