The Bahadur-Kiefer representation of L(p) regression estimators

We consider the following linear regression model: Y-i = Z(i)'theta(0) + U-i, i = 1,...,n, where X(1) = (Y-1', Z(1)),...,X(n) = (Y-n', Z(n)) are independent and identically distributed random variables, Y-i is real, Z(i) has values in R(m), U-i is independent of Z(i), and theta(0) is and m-dimensional parameter to be estimated. The L(p) estimator of theta(0) is the value theta(n) such that n(-1) (i=1)Sigma(n) \Y-i - Z(i)'theta(n)\(p) = (theta is an element of Rm) inf n(-1) (i=1)Sigma(n) \Y-i - Z(i)'theta\(p). Here, we will give the extract Bahadur-Kiefer representation of theta(n), for each p greater than or equal to 1. Explicitly, we will see that, under regularity conditions, (i) if p > 3/2, (n-->infinity)lim sup (n/log log n)(log n)\H'(theta(0)).(theta(n)-theta(0)) + H-n(theta(0))\ = c a.s., (ii) if p = 3/2, (n-->infinity)lim sup (n/log log n)(log n)(-1/2)\H'(theta(0)).(theta(n)-theta(0)) + H-n(theta(0))\ = c a.s., (iii) if 3/2 > p greater than or equal to 1, (n-->infinity)lim sup (n/log log n)((2p+1)/4)\H'(theta(0)).(theta(nt)heta 0) + H-n(theta(0))\ = c a.s., where h(x,theta) = sign (y - z'theta)\y - z'theta)\y - z'theta\(p-1) z, H(theta) = E[h(X,theta)], H-n(theta) = n(-1)Sigma(i=1)(n) h(X(i), theta) and c is a positive constant, which depends on p and on the random variable X.