在相关偏差情况下,线性回归模型较高系数的ls和Aitken估计的相等性

IF 0.1
M. Savkina
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引用次数: 1

摘要

本文研究了函数形式为$f(x) = ax + b$, $a$和$b$ -未知参数的线性回归模型。函数$f(x)$的近似值(观测值)在等距点$x_0$, $x_1$,…, $x_n$为线段。并假设偏差的协方差矩阵为Toeplitz矩阵。在所有Toeplitz矩阵中,选择一组矩阵,其中从(k +1)-th开始,所有平行于主矩阵的对角线均为零,$k = n/2$, $n$ -偶数。主对角线上的元素记为$λ$,第k条对角线上的元素记为$c$,第j条对角线上的元素记为$c_{k−j}$, $j = 1,2,…, k−1$。本文证明了如果$c_j = (k/(k + 1))^ jc $, $j = 1,2,…, k−1$,使得该模型参数$a$的LS估计和Aitken估计对于任意值$λ$和$c$重合,从而提供了结果矩阵的正确定性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
EQUALITY OF LS AND AITKEN ESTIMATIONS OF THE HIGHER COEFFICIENT OF THE LINEAR REGRESSION MODEL IN THE CASE OF CORRELATED DEVIATIONS
At the paper a linear regression model whose function has the form $f(x) = ax + b$, $a$ and $b$ — unknown parameters, is studied. Approximate values (observations) of functions $f(x)$ are registered at equidistant points $x_0$, $x_1$,..., $x_n$ of a line segment. It is also assumed that the covariance matrix of deviations is the Toeplitz matrix. Among all Toeplitz matrices, a family of matrices is selected for which all diagonals parallel to the main, starting from the (k +1)-th, are zero, $k = n/2$, $n$ — even. Elements of the main diagonal are denoted by $λ$, elements of the k-th diagonal are denoted by $c$, elements of the j-th diagonal are denoted by $c_{k−j}$ , $j = 1, 2,..., k − 1$. The theorem proved at the paper states that if $c_j = (k/(k + 1))^j c$, $j = 1, 2,..., k−1$, that the LS estimation and the Aitken estimation of the $a$ parameter of this model coincide for any values $λ$ and $c$, which provide the positive definiteness of the resulting matrix.
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