Generalized Levinson-Durbin sequences, binomial coefficients and autoregressive estimation
Generalized Levinson-Durbin sequences, binomial coefficients and autoregressive estimation
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广义 Levinson-Durbin 序列、二项式系数和自回归估计
DOI：
10.1016/j.jmva.2010.01.004
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发表时间：
2010
期刊：
影响因子：
--
通讯作者：
P. Shaman
中科院分区：
文献类型：
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作者：
P. Shaman
对于离散时间二阶平稳过程，在给定k个先前观测值的情况下，利用Levinson-Durbin递推确定k+1时刻观测值的最佳线性预报器的系数，在最小化均方误差的意义下是最好的。由递归确定的系数定义了Levinson-Durbin序列。我们还定义了广义Levinson-Durbin序列，并注意到二项式系数形成广义Levinson-Durbin序列的特例。证明了所有广义Levinson-Durbin序列都服从求和公式，推广了二项式系数满足的公式。Levinson-Durbin序列是在几个自回归模型系数估计的构造过程中产生的。最小二乘自回归估计器不产生Levinson-Durbin序列，但最小二乘不动点过程可以组合以构造Levinson-Durbin序列，其中T是样本长度，其中，最小二乘不动点过程产生1/T阶无偏系数的最小二乘估计。相反，由Yule-Walker估计器产生的类似的不动点过程并不组合来构造Levinson-Durbin序列，尽管Yule-Walker估计器本身确实确定Levinson-Durbin序列。进一步研究了均值为多项式时间趋势的最小二乘过程和Yule-Walker不动点过程。
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 1/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.