Asymptotic distribution of regression M-estimators

We consider the following linear regression model: Y-i = Z(i)' theta (0) + U-i, i = 1, ..., n, where {U-i}(i=1)(infinity) is a sequence of R-m-valued i.i.d. r.v.'s; {Z(i)}(i=1)(infinity) is a sequence of i.i.d. d x m random matrices; and theta (0) is a d-dimensional parameter to be estimated. Given a function rho : R-m --> R, we define a robust estimator <(<theta>)over cap>(n), as a value such that n(-1) Sigma (n)(i=1) rho (Y-i - Z(i)'<(<theta>)over cap>(n)) = inf(theta is an element of Rd) n(-1) Sigma (n)(i=1) rho (Y-i - Z(i)' theta). We study the convergence in distribution of a(n)(<(<theta>)over cap>(n) - theta (0)) in different situations, where {a(n)} is a sequence of real numbers depending on rho and on the distributions of Z(i) and U-i. As a particular case, we consider the case rho (x)= \x \ (p). In this case, we show that if E[parallel toZ parallel to (p) + parallel toZ parallel to (2)] < infinity; either p > 1/2 or m greater than or equal to 2; and same other regularity conditions hold, then n(1/2)(<(<theta>)over cap>(n) - theta (0)) converges in distribution to a normal limit. For m = 1 and p = 1/2, n(1/2)(log n)(-1/2)(<(<theta>)over cap>(n) - theta (0)) converges in distribution to a normal limit. For m = 1 and 1/2 > p > 0, n(1/(3 - 2p))(<(<theta>)over cap>(n) - theta (0)) converges in distribution. (C) 2001 Elsevier Science B.V. All rights reserved.