Kalman filter, outliers, missing observations, time series analysis, 93E20, 62F35,

The paper is devoted to robust modifications of exponential smoothing for time series with outliers or long-tailed distributions. Classical exponential smoothing applied to such time series is sensitive to the presence of outliers or long-tailed distributions and may give inadequate smoothing and forecasting results. First, simple and double exponential smoothing in the L1 norm (i.e. based on the least absolute deviations) are discussed in detail. Then, general exponential smoothing is made robust, replacing the least squares approach by M-estimation in such a way that the recursive character of the final formulas is preserved. The paper gives simple algorithmic procedures which preserve advantageous features of classical exponential smoothing and, in addition, which are less sensitive to outliers. Robust versions are compared numerically with classical ones. KEY WORDS Exponential smoothing Outliers Long-tailed distributions Robust methods L1 norm Least absolute deviations M-estimation Time seriesExponential smoothing is a popular recursive smoothing and prediction technique frequently used for routine treatment of time series. Despite its numerical simplicity, it gives good practical results in comparison with more complicated smoothing and forecasting methods (see e.g. Gardner, 1985). Although exponential smoothing has an adaptive character due to the exponential weighting of observations in the past, its results may be distorted by the presence of outliers in time series. For instance, Ledolter (1989) has investigated the effect of outliers on the forecasts in ARIMA models, the connection between these models and exponential smoothing being known (e.g. between the ARIMA(O,l, 1) model and simple exponential smoothing). Outliers are frequently explained as a contamination of time series by long-tailed distributions according to various models (e.g. so-called additive and innovation outliers), and there exist corresponding methods of robust time-series analysis suitable for particular methods (see e.g. Tsay, 1988, or the survey paper by Stockinger and Dutter, 1987). In the framework of exponential smoothing, indirect methods of outliers treatment consisting of identification of outliers and interpolation of missing values have been preferred (see e.g. Aldrin and Damsleth, 1989;Cipra, 1989; Jun, 1989). This paper suggests direct approaches to making the exponential smoothing more robust in order to become less sensitive to outliers.First, exponential smoothing in the L I norm is suggested, since this norm is known to be

The paper solves the loss reserving problem using Kalman recursions in linear statespace models. In particular, if one orders claims data from run-off triangles to time series with missing observations, then state space formulation can be applied for projections or interpolations of IBNR (Incurred But Not Reported) reserves. Namely, outputs of the corresponding Kalman recursion algorithms for missing or future observations can be taken as the IBNR projections. In particular, by means of such recursive procedures one can perform effectively simulations in order to estimate numerically the distribution of IBNR claims which may be very useful in terms of setting and/or monitoring of prudency level of loss reserves. Moreover, one can generalize this approach to the multivariate case of several dependent run-off triangles for correlated business lines and the outliers in claims data can be also treated effectively in this way. Results of a numerical study for several sets of claims data (univariate and multivariate ones) are presented.

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The paper presents a simple procedure for interpolating, smoothing, and predicting in seasonal time series with missing observations. The approach suggested by Wright (Wright, D. J. 1986. Forecasting data published at irregular time intervals using extension of Holt's method. Management Sci. 32 499--510.) for the Holt's method with nonseasonal data published at irregular time intervals is extended to the Holt-Winters method in the seasonal case. Numerical examples demonstrate the procedure.missing observations, seasonality, Holt-Winters method