论文标题
住户遇到学生
Householder Meets Student
论文作者
论文摘要
用于高的薄n x P全级矩阵X的QR分解的住户算法具有产生带有正顺序柱的矩阵M的额外好处，这是X柱空间的矫形器的基础。当x在该正排出时，我们给出了M-Transpose X的简单公式。该公式不需要计算M，仅需要QR分解的R因子。这用于获得经典线性回归中独立“残差”的非常简单的可计算混凝土表示。对于学生问题，当p = 1时，如果r（j）= y（j）-ybar是通常的（非独立）残差，则W（j）= r（j+1）-r（1）/（sqrt（n）+1）给出n-1 i.i.d.平均正态变量的平均变量与N残差相同。该公式的这些特性（事后看）可以很容易地直接验证，从而获得了学生定理的新简单而具体的证明。它还提供了一种简单的方法来生成N-1正好均值零i.i.d.来自n样品的样本未知。 Yiping Cheng在学生定理的建设性证明的背景下，表现出Y（J）与这些属性的具体线性组合，但是这种表示并不是那么简单。当有更多的预测因子时，将获得回归的类似简单结果，为N-P I.I.D提供了非常简单的可计算混凝土公式。具有与通常的n非独立残差相同的平方和平方和平方的独立残差。讨论了与科克伦定理的联系。
The Householder algorithm for the QR factorization of a tall thin n x p full-rank matrix X has the added bonus of producing a matrix M with orthonormal columns that are a basis for the orthocomplement of the column space of X. We give a simple formula for M-Transpose x when x is in that orthocomplement. The formula does not require computing M, it only requires the R factor of a QR factorization. This is used to get a remarkably simple computable concrete representation of independent "residuals" in classical linear regression. For Students problem, when p=1, if R(j)=Y(j)-Ybar are the usual (non-independent) residuals, W(j)=R(j+1) - R(1)/(sqrt(n)+1) gives n-1 i.i.d. mean-zero normal variables whose sum of squares is the same as that of the n residuals. Those properties of this formula can (in hindsight) easily be verified directly, yielding a new simple and concrete proof of Student's theorem. It also gives a simple way of generating n-1 exactly mean-zero i.i.d. samples from n samples with unknown mean. Yiping Cheng exhibited concrete linear combinations of the Y(j) with these properties, in the context of a constructive proof of Student's theorem, but that representation is not so simple. Analogous simple results are obtained for regression when there are more predictors, giving a very simple computable concrete formula for n-p i.i.d. independent residuals with the same sum of squares as that of the usual n non-independent residuals. A connection with Cochran's theorem is discussed.
