It is well-known in the theory of stationary stochastic process) that, given the true spectrum and subject to general conditions, the linear least squares predictor (i.e., extrapolator) has an error variance not smaller than the error variance of the linear least squares interpolator. It is then perhaps natural to expect that the same results would hold when the true spectrum is unknown, but can be estimated from data. In this correspondence the fallacy of this expectation is exposed by carefully analyzing a simple situation. The seemingly paradoxical phenomenon is further demonstrated by some simulation results. We conclude with a heuristic explanation.

Tong has proposed an objective method, based on Akaike's information criterion (AIC), for the determination of the order of an autoregressive (AR) model with noisy data; some extensions of this proposed method are discussed.

Recently, Tong has studied the problem of fitting an autoregressive (AR) model to a stochastic signal with noisy data. In this correspondence, we study the associated problem of fitting a smooth moving average to noisy data. The approach is again based on an information criterion due to Akaike.

Davisson [131, [141 has considered the problem of determining the "order" of the signal from noisy data. Although interesting theoretically, his result is difficult to use in practice. In this correspondence, we exploit one well-known fact concerning autoregressive (AR) signals plus white noise, and using Akaike's information criterion [15], [17], we have developed one efficient procedure for determining the order of the AR signal from noisy data. The procedure is illustrated numerically using both artificially generated and real data. The connection between the preceding problem and the classical statistical problem of factor analysis is discussed.