CONDITIONING VS. STANDARDIZATION FOR CONTRASTS ON ERRORS ATTRACTED TO STABLE LAWS

Abstract

In multiple regression and other settings one encounters the problem of estimating sampling distributions for contrast operations applied to i.i.d. errors. Permutation bootstrap applied to least squares residuals has been proven to consistently estimate conditionalsampling distributions of contrasts, conditional upon order statistics of errors, even for long-tailed error distributions. How does this compare with the unconditional sampling distribution of the contrast when standardizing by the sample s.d. of the errors (or the residuals)? For errors belonging to the domain of attraction of a normal we present a limit theorem proving that these distributions are far closer to one another than they are to the limiting standard normal distribution. For errors attracted to α-stable laws with α ≤ 2 we construct random variables possessing these conditional and unconditional sampling distributions and develop a Poisson representation for their a.s. limit correlation ρ_α. We prove that ρ₂= 1, ρ_α→ 1 for α → 0 + or 2 −, and ρ_α< 1 a.s. for α < 2.

Keywords:

• Domain of stable attraction
• Multiple regression
• Conditional bootstrap
• Contrast vector

Acknowledgments