ANALYTICAL AND BOOTSTRAP APPROXIMATIONS TO ESTIMATOR DISTRIBUTIONS INL(1) REGRESSION

Edgeworth and bootstrap approximations to estimator distributions in L 1 regression are described. Analytic approximations based on Edgeworth expansions that mix lattice and nonlattice components and allow for a n intercept term in the regression are developed under mild conditions , which do not even require a density for the error distribution. Unde r stronger assumptions on the error distribution, the Edgeworth expans ion assumes a simpler form. Bootstrap approximations are described, an d the consistency of the bootstrap in the L1 regression setting is est ablished. We show how the slow rate n-1/4 of convergence in this conte xt of the standard, unsmoothed bootstrap that resamples for the raw re siduals may be improved to rate n-2/5 by two methods. a smoothed boots trap approach based on resampling from an appropriate kernel estimator of the error density and a normal approximation that uses a kernel es timator of the error density at a particular point, its median 0. Both of these methods require choice of a smoothing bandwidth, however. Nu merical illustrations of the comparative performances of the different estimators in small samples are given, and simple but effective empir ical rules for choice of smoothing bandwidth are suggested.