Generalized Levinson-Durbin sequences, binomial coefficients and autoregressive estimation

JOURNAL OF MULTIVARIATE ANALYSIS

卷:101

Article

Shaman, Paul

关键词: Binomial coefficients;    Generalized Levinson-Durbin sequence;    Least squares estimator;    Levinson-Durbin sequence;    Partial correlations;    Yule-Walker estimator;   

DOI  :  10.1016/j.jmva.2010.01.004

来源: Elsevier

【 摘 要 】

For a discrete time second-order stationary Process, the Levinson-Durbin recursion is used to determine the coefficients of the best linear predictor of the observation at time k + 1, given k previous observations, best in the sense of minimizing the mean square error. The coefficients determined by the recursion define a Levinson-Durbin sequence. We also define a generalized Levinson-Durbin sequence and note that binomial coefficients form a special case of a generalized Levinson-Durbin sequence. All generalized Levinson-Durbin sequences are shown to obey summation formulas which generalize formulas satisfied by binomial coefficients. Levinson-Durbin sequences arise in the construction of several autoregressive model coefficient estimators. The least squares autoregressive estimator does not give rise to a Levinson-Durbin sequence, but least squares fixed point processes, which yield least squares estimates of the coefficients unbiased to order I/T, where T is the sample length, can be combined to construct a Levinson-Durbin sequence. By contrast, analogous fixed point processes arising from the Yule-Walker estimator do not combine to construct a Levinson-Durbin sequence, although the Yule-Walker estimator itself does determine a Levinson-Durbin sequence. The least squares and Yule-Walker fixed point processes are further studied when the mean of the process is a polynomial time trend that is estimated by least squares. (C) 2010 Elsevier Inc. All rights reserved.

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