Small sample behavior of robust stochastic approximation and iterated weighted least squares estimates for location

A class of robuet estimates of location uses the Robbins-Monro stochastic approximation algorithm as a basis. These estimates, called SA-estimates, have been proposed by Martin (1972), who established an asymptotic min-max result like that of P. Huber (1964) for M-estimates. Here we study in detail the small sample behavior of robust SA-estimates for location at sample sizes N = 10, 20, 40 using Monte Carlo swindle techniques. Results are presented for point estimate efficiencies and for error rates and expected confidence interval lengths obtained by studentizing through use of the natural estimate of the asymptotic variance formula. Unlike M-estimates, SA-estimates are not invariant under permutations of the data order. Thus our study included one-step M-estimates and iterated- weighted least squares estimates. This was done not only for comparison purposes, but also because the latter are of interest in their own right. The results show that SA-estimate losses due to the lack of invariance under permutation of the data order are small, and that the performance of various SA-estimates compares favorably with that of M-estimates.