A data analytic approach to the smoothing problem and some of its variations

Let y(n) = f(n) + ε(n), n=1,...,N with the ε(n) i.i.d, from ε(n) ~ N(0, σ2), σ2 unknown and f(·) an unknown "smooth function". The problem is to estimate f(n), n=1,...,N in some way that makes sense statistically. E.T. Whittaker, 1919, suggested that the solution have the property that it be a trade off between fidelity to the data and fidelity to a difference equation constraint. Craven and Wahba, 1979 is an 0(N3) solution to the problem, Akaike, 1979 is an 0(N2) solution to this problem. The Whittaker, Wahba and Akaike solutions are described. In our approach alternative candidate "smooth" models are imbedded into dynamic state space forms. The recursive computational Kalman smoother procedure is invoked to achieve an 0(N) computation. Akaike's AIC, a statistical decision procedure, is employed to determine the best of alternative Kalman filter-predictor modeled data. The Kalman smoother solution corresponding the best AIC Kalman filter, is then the best (fixed interval) smooth solution of the data. The smoothing problem is generalized to consider the smoothing of econometric time series with trends and seasonalities, trends with correlated noise and smooth functions with outliers. Each of these variations of the smoothing problem is "solved" by an iterative data analytic approach. In that approach the results of tentative first analyses suggest new candidate dynamic constraint models, and in the case of robust smoothing, a new procedure. Some of the final AIC criterion results are seemingly surprising and counterintuitive. Examples are shown.