Optimized estimation of moments for nonstationary signals

The behavior of nonstationary moment estimators that are based on linear time-invariant filtering is analyzed. The performance of such estimators is evaluated in terms of their time-averaged variance and time-averaged squared bias. Optimal estimators that minimize a convex combination of bias and variance are derived. The superiority of such optimally weighted filtering over the conventional (exponentially windowed) moment estimation technique is demonstrated by means of a simple example. The same example also serves to illustrate the difficulties encountered when the construction of optimal estimators relies on uncertain prior information, as well as to demonstrate the feasibility of overcoming such difficulties by using appropriately designed (robust) optimal estimators. We also include a brief discussion of the relevance of the proposed moment estimation technique to identification of linear time-variant systems and to estimation of time-variant spectra.