On the Asymptotic Distribution of Exponentially Weighted Prediction Error Estimators

This correspondence establishes the distribution of the exponentially weighted prediction error estimators for a class of general (possibly nonlinear) discrete-time systems. An explicit formula for the covariance matrix of the estimation errors is provided. The distributional results presented hold for a large number N of data points and a weighting (or forgetting) factor λ close to one. The covariance matrices corresponding to {N1, >> 0; λ1, = 1} and to {N2 >>> 0; λ2 = 1 - δ, where δ is a small positive number} are shown to be related in a simple way. Specifically, it is shown that they are equal iff N1, = 2/(1 - λ2). This extends a result recently obtained by Porat in the special case of times-series autoregressive models.