Symmetry, Regression Design, and Sampling Distributions
研究发现当回归变量对称分布时，许多M估计量的抽样分布具有反射性质，即系数符号反转时分布关于原点对称，这有助于设计蒙特卡洛实验和解释估计量性质。
When values of regressors are symmetrically disposed, many M -estimators in a wide class of models have a reflection property, namely, that as the signs of the coefficients on regressors are reversed, their estimators' sampling distribution is reflected about the origin. When the coefficients are zero, sign reversal can have no effect. So in this case, the sampling distribution of regression coefficient estimators is symmetric about zero, the estimators are median unbiased and, when moments exist, the estimators are exactly uncorrelated with estimators of other parameters. The result is unusual in that it does not require response variates to have symmetric conditional distributions. It demonstrates the potential importance of covariate design in determining the distributions of estimators, and it is useful in designing and interpreting Monte Carlo experiments. The result is illustrated by a Monte Carlo experiment in which maximum likelihood and symmetrically censored least-squares estimators are calculated for small samples from a censored normal linear regression, Tobit, model.