The paper considers a system described by the state-space model \[ x_ n=F x_{n-1}+G v_ n,\quad y_ n=H x_ n+w_ n, \] where F, G, H are linear transformations R \(k\to R\) k, \(R^{\ell}\to R\) K, R \(k\to R\), respectively, and \(v_ n\), \(w_ n\) are independent, not necessarily Gaussian random variables. Here formulas are derived for one-step-ahead prediction, filtering, smoothing, joint density of \(x_ n\) and \(x_{n- 1}\), given the entire observation sequence \((y_ 1,...,y_ N)\). There are several interesting examples of artificially generated and real data, where the non-Gaussian model “explains” the large deviation by having a heavier tail rather than shifting the distribution itself as well as multimodal or skewed distributions, jump of the mean and other strange phenomena.
The paper is supplied with several comments on the aforementioned and related subjects such as application of non-Gaussian filtering to nonlinear models (Kohn, Ansley), discussion of robust procedures for mixtures of two Gaussian density models (Martin, Raftery, O’Sullivan, Tsay), irregularly spaced data (Wahba). The paper also contains the author’s reply to the comments.