We survey the asymptotic properties of regression [L.sub.p] estimators under general classes of error distributions. It is found that the asymptotic distributions of [L.sub.p] estimators depend crucially on p and the shape of the error distribution near the origin. A number of important features arise as a result, among which are (a) use of a small p may yield accelerated convergence rates for [L.sub.p] estimators under certain classes of error distributions; (b) for p

KEY WORDS: Convergence rate; Gaussian process; [L.sub.p] estimator; m out of n bootstrap; Regression.

1. INTRODUCTION

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<2, 2p + [zeta]><2, 2p + [zeta]><0, for some [[zeta].sup.+], [[zeta].sup.-]><1 and [F'.sub.U](0)><1 and p><0 or><0 and [[gamma].sub.0]><1 and [zeta]><1 and [zeta]><1. Figure 7 summarizes the results, which show that the smallest MSE is again achieved at p around 1.5 in all cases. Larger values of p hamper the accuracy due to their sensitivity to outliers. We observe that the MSE decreases slightly when p decreases below 1, and the trend is unaffected by outliers. In all cases, [L.sub.1] estimates are the least accurate, reflecting their slow convergence rate, [n.sup.1/(2[zeta])] for [zeta]><1 but slower than [n.sup.1/2] when [zeta]><1/2. Setting p [member of] (1/2, 1] actually gives an optimal convergence rate in the sense that it is comparable to that of any asymptotically efficient estimator of [[beta].sub.0]. Thus there are circumstances under which use of a small p not only guards against outliers or heavy-tailed errors, but also provides a means of constructing estimators that yield optimal convergence rates, especially when [F.sub.U] is heavy-tailed or varies regularly at a fast rate at 0. Such ratewise optimality decidedly warrants an in-depth investigation into the asymptotic properties and practical implementation of [L.sub.p] estimation for small p. Our simulation study provides empirical evidence that both robustness (against outliers), and efficiency can be achieved simultaneously if the error distribution has heavy tails and a central shape that favors using a small p for faster convergence. Using a large p (><[r.sub.n]||[beta] - [[beta].sub.0]|| [less than or equal to] [2.sup.j]}. For fixed [eta], M><1 and q>

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