The change in the least squares estimator (LSE) of a vector of regression coefficients due to a case deletion is often used for investigating the influence of an observation on the LSE. A normalization of the change in the LSE using the Moore-Penrose inverse of the covariance matrix of the change in the LSE is derived. This normalization turns out to be a square of the internally studentized residual. It is shown that the numerator term of Cook's distance does not in general have a chi-squared distribution except for a single case. An elaborate explanation about the inappropriateness of the choice of a scaling matrix defining Cook's distance is given. By reflecting a distributional property of the change in the LSE due to a case deletion, a new diagnostic measure that is a scalar is suggested. Three numerical examples are given for illustration.
翻译:由于案例删除而使回归系数矢量最小方位估计值(LSE)的变化,通常用于调查对 LSE观察的影响。使用 Moore-Penrose 来代替LSE变化的共变矩阵,得出LSE变化的正常化。这种正常化是内部学习剩余量的平方。显示除了一个案例外,Cook距离的点数术语一般没有奇差分布。对确定Cook距离的缩放矩阵选择不适当作了详细解释。通过反映LSE因案件删除而变化的分布属性,建议了一个新的诊断性计量尺度。为说明提供了三个数字示例。