The ROBUSTREG Procedure

Details: ROBUSTREG Procedure

This section describes the statistical and computational aspects of the ROBUSTREG procedure. The following notation is used throughout this section.

Let $ \bX = (x_{ij})$ denote an $n\times p$ matrix, $ \mb {y}=(y_1,\ldots ,y_ n)’$ denote a given n-vector of responses, and $\btheta = (\theta _1,\ldots ,\theta _ p)’$ denote an unknown p-vector of parameters or coefficients whose components are to be estimated. The matrix $\bX $ is called the design matrix. Consider the usual linear model

\[  \mb {y} = \bX \btheta + \mb {e}  \]

where $ \mb {e} = (e_1,\ldots ,e_ n)’$ is an n-vector of unknown errors. It is assumed that (for a given $\bX $) the components $e_ i$ of $\mb {e}$ are independent and identically distributed according to a distribution $L(\cdot / \sigma )$, where $\sigma $ is a scale parameter (usually unknown). Often $L(\cdot ) \approx \Phi (\cdot )$, the standard normal distribution function. The vector of residuals for a given value of $\hat\btheta $ is denoted by $\mb {r} = (r_1,\ldots ,r_ n)’$ and the ith row of the matrix $\bX $ is denoted by $ \mb {x}_ i’$.