A Computational Method For Estimating Densities Of Non‐Gaussian Nonstationary Univariate Time Series

. Kitagawa's (1987a) numerical integration formulae used to approximate the filtering, prediction and smoothing densities of nonlinear non‐Gaussian state‐space models are modified. The method involves integration by parts which permits the integration of the conditional system density and, possibly, the observational density prior to and independently of the computation of the filtering, prediction and smoothing densities. In addition to a substantial reduction in computing time and an increase in accuracy, this approach eliminates the necessity of incorporating dynamic adjustments to the filtering, prediction and smoothing processes to accommodate difficult noise densities. Three numerical examples are presented. One example replicates Kitagawa's non‐Gaussian state‐space model; the second is a linear Gaussian model; and the third is a nonlinear non‐Gaussian model. Comparisons of speed and accuracy between alternative methods and between three computers (personal computer, minicomputer and supercomputer) are made.