The filtering of time series with unknown signal statistics

The optimum operation for the smoothing of time series to minimize mean square error is well known when the signal and noise statistics are completely available. When the statistics of either or both are partially or completely unknown, however, there exists no universally agreed upon optimum procedure. This paper considers the problem of linear smoothing when the noise is additive, signal independent with first and second moments known while the signal statistics are completely unknown. This case is of practical interest since frequently signal assumptions are difficult to make whereas the noise can usually be assumed to be sample-to-sample uncorrelated with mean zero and known variance. This latter assumption of known noise power can be relaxed in some instances as will be discussed in a future paper.