# On Ranks of Regression Errors and Residuals

Authors
Type
Preprint
Publication Date
Feb 27, 2012
Submission Date
Feb 09, 2012
Identifiers
arXiv ID: 1202.1964
Source
arXiv
Consider the standard linear regression model $Y = D \theta + \epsilon$ with given design matrix $D$ ($n \times p$), unknown parameter $\theta$ ($p \times 1$) and unobserved error vector $\epsilon$ ($n \times 1$) with i.i.d.\ centered Gaussian components. Motivated by an application in economics, we compare the ranks $R_i$ of the errors $\epsilon_i$ with the ranks $\hat{R}_i$ of the residuals $\hat{\epsilon}_i$, where $\hat{\epsilon} = Y - D \hat{\theta}$ with the least squares estimator $\hat{\theta}$. Exact and approximate formulae are given for the rank distortions $\sqrt{E \, (\hat{R}_i - R_i)^2}$.