The Monitoring of Exponentially Weighted Forecasts

By considering equivalences between various forecasting systems, the exact stochastic process followed by the one-step-ahead errors may be found. This process, the error process, is important for any monitoring scheme, and is a function of the forecasting system and the underlying data process. The error process is obtained for the most general form of exponential smoothing systems used in optimal conditions. The statistical properties are derived. In particular, the approximate variance of Trigg's smoothed error tracking signal is obtained explicitly for several exponential smoothing systems, and a procedure is given for obtaining it numerically for any such system. The use of different smoothing constants in the forecasting system and the tracking signal is discussed and it is found that suitable choice of the tracking signal constant greatly improves the performance of the signal, making it more comparable with CUSUM schemes.