THE TWO-FILTER FORMULA FOR SMOOTHING AND AN IMPLEMENTATION OF THE GAUSSIAN-SUM SMOOTHER

GENSHIRO KITAGAWA

The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, Tokyo 106, Japan

(Received January 4, 1993; revised February 4, 1994)

Abstract. A Gaussian-sum smoother is developed based on the two filter formula for smoothing. This facilitates the application of non-Gaussian state space modeling to diverse problems in time series analysis. It is especially useful when a higher order state vector is required and the application of the non-Gaussian smoother based on direct numerical computation is impractical. In particular, applications to the non-Gaussian seasonal adjustment of economic time series and to the modeling of seasonal time series with several outliers are shown.

Key words and phrases: Non-Gaussian smoother, non-Gaussian filter, Gaussian mixture, nonstationary time series, outliers, seasonal adjustment.