Minimax-robust prediction of discrete time series

SummaryWe discuss a robust approach for predicting a weakly stationary discrete time series whose spectral density f is not exactly known. We assume that we know that f∈ $$\mathfrak{D}$$ ), where $$\mathfrak{D}$$ is a convex set of spectral densities fulfilling some not too stringent conditions. We proof the existence of a “most indeterministic” density f0 in $$\mathfrak{D}$$ , and we show that the classical optimal linear predictor for a time series with spectral density f0 is mini-max-robust in the sense that it minimizes the maximal possible prediction error.We investigate some special models $$\mathfrak{D}$$ , and, in doing so, we illustrate a generally applicable method for determining the robust predictor. In particular, we discuss model sets $$\mathfrak{D}$$ which are defined by conditions on a finite part of the autocovariance sequence of the corresponding time series. These examples are of particular interest as the most indeterministic density is an autoregressive one, i.e. the robust predictor is finite. We discuss connections between this type of model set $$\mathfrak{D}$$ and maximum entropy and generalized maximum entropy spectral estimates.