LINEAR REPRESENTATION OF M-ESTIMATES IN LINEAR-MODELS

Consider the linear regression model, y(i) = x(i)(T)beta0 + e(i), i = 1, ..., n, and an M-estimate beta of beta0 obtained by minimizing SIGMArho(y(i) - x(i)(T)beta), where rho is a convex function. Let S(n) = SIGMAx(i)x(i)T and r(n) = S(n)1/2(beta - beta0) - S(n)-1/2 SIGMAx(i)h(e(i)), where, with a suitable choice of h(.), the expression SIGMAx(i)h(e(i)) provides a linear representation of beta. Bahadur (1966) obtained the order of r(n) as n --> infinity when beta0 is a one-dimensional location parameter representing the median, and Babu (1989) proved a similar result for the general regression parameter estimated by the LAD (least absolute deviations) method. We obtain the stochastic order of r(n) as n --> infinity for a general M-estimate as defined above, which agrees with the results of Bahadur and Babu in the special cases considered by them.