Approximating the internal norm influence measure in linear regression

Abstract

The internal norm measure D_I(C⁻¹ _k) introduced in Gray (1989) assesses the influencethat a size- subset of cases has on the least squares parameter estimates in a linear regression. The term internal norm refers to the fact that the assessment of influence is based on a comparison of the subset of interest to all size- subsets internal to the data set. Although this approachis intuitively

appealing; it can be impractical for k > 1 since the calculation of D_I(C⁻¹ _k) for a size-k subset I requires the covariance matrix C_k of all size-A' deletion parameter estimates. Using results from finite population sampling theory, we show that the covariance matrices C_k and C_I are nearly proponional for large sample sizes, a fact

that can be used to inexpensively approximate D_I(C⁻¹ _k) for subsets containing two or more cases. Numerical evidence from several real data sets indicates that the approximation is highly accurate for large data sets, where it is most needed, and sufficiently accurate for small to moderate regression problems.

Keywords:

• regression diagnostics
• influential data
• ratio estimator
• leverage
• elemental regression