Generalized Levinson-Durbin sequences, binomial coefficients and autoregressive estimation

For { t y } a discrete time, second-order stationary process, the Levinson-Durbin recursion is used to determine the coefficients , jk α j=1, … , k, of the best linear predictor of 1 + k y , , ˆ 1 1 1 y y y kk k k k α α − − − = + L best in the sense of minimizing the mean square error. The coefficients jk α determine a Levinson-Durbin sequence. A generalized Levinson-Durbin sequence, a special case of a sequence of generalized binomial coefficients, is studied. Binomial coefficients form a generalized Levinson-Durbin sequence, and all generalized Levinson-Durbin sequences are shown to obey some summation formulas which generalize summations satisfied by binomial coefficients. The summation formulas are expressed in terms of the partial correlation sequence. Levinson-Durbin sequences arise in the construction of autoregressive model coefficient estimates for the Yule-Walker, tapered Yule-Walker, Burg and Kay estimators. The least squares autoregressive estimator, though, does not give rise to a Levinson-Durbin sequence. However, least squares fixed point processes, which yield least squares estimates of the coefficients unbiased to order 1/T, where T is the sample length, can be combined to construct Levinson-Durbin sequences. By contrast, analogous Yule-Walker fixed point processes do not combine to construct Levinson-Durbin sequences. The least squares fixed point processes are studied when the mean of the process is a polynomial time trend that is estimated by least squares. For each degree of polynomial trend, the fixed point processes form a sequence of projections from an infinite order fixed point process. The correlation functions and spectral 2 densities of these infinite order fixed point processes are derived for polynomial trends of degree up to 5.