Analytical and bootstrap approximations to estimator distributions in L1 regression

D. De Angelis, Peter Hall, G. A. Young

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    39 Citations (Scopus)

    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 L1 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.

    Original languageEnglish
    Pages (from-to)1310-1316
    Number of pages7
    JournalJournal of the American Statistical Association
    Volume88
    Issue number424
    DOIs
    Publication statusPublished - Dec 1993

    Keywords

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

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