Robust estimation via stochastic approximation

It has been found that robust estimation of parameters may be obtained via recursive Robbins-Monro-type stochastic approximation (SA) algorithms. For the simple problem of estimating location, appropriate choices for the nonlinear transformation and gain constant of the algorithm lead to an asymptotically min-max robust estimator with respect to a family
\mathcal{F} (y_p,p)
of symmetrical distributions having the same mass
p
outside
[-y_p,y_p], 0 < p < 1
. This estimator, referred to as the
p
-point estimator (PPE), has the additional striking property that the asymptotic variance is constant over the family
\mathcal{F}(Y_p,p)
. The PPE is also efficiency robust in large samples. Monte Carlo results indicate that small sample robustness may be obtained using both one-stage and two-stage procedures. The good small-sample results are obtained in the one-stage procedure by using an adaptive gain sequence, which is intuitively appealing as well as theoretically justifiable. Some possible extension of the SA approach are given for the problem of estimating a vector parameter. In addition, some aspects of the relationship between SA-type estimators and Huber's
M
-estimators are given.