Analytical and Bootstrap Approximations to Estimator Distributions in L¹ Regression

Abstract

Edgeworth and bootstrap approximations to estimator distributions in L¹ regression are described. Analytic approximations based on Edgeworth expansions that mix lattice and nonlattice components and allow for an intercept term in the regression are developed under mild conditions, which do not even require a density for the error distribution. Under stronger assumptions on the error distribution, the Edgeworth expansion assumes a simpler form. Bootstrap approximations are described, and the consistency of the bootstrap in the L ¹ regression setting is established. We show how the slow rate n ^−1/4 of convergence in this context of the standard, unsmoothed bootstrap that resamples for the raw residuals may be improved to rate n ^−2/5 by two methods: a smoothed bootstrap approach based on resampling from an appropriate kernel estimator of the error density and a normal approximation that uses a kernel estimator of the error density at a particular point, its median 0. Both of these methods require choice of a smoothing bandwidth, however. Numerical illustrations of the comparative performances of the different estimators in small samples are given, and simple but effective empirical rules for choice of smoothing bandwidth are suggested.

Key Words:

• Density estimator
• Edgeworth expansion
• L ¹ regression
• Normal approximation
• Smoothed bootstrap