Efficiency-Constrained Bias-Robust Estimation of Location

Abstract

In 1964, P. Huber established the following minimax bias robustness result for estimating the location $\mu$ in the $\varepsilon$-contamination family $F(x)=(1-\varepsilon)\phi[(x-\mu)/s]+\varepsilon H(x)$, where $\phi$ is the standard normal distribution and H is an arbitrary distribution function: The median minimizes the maximum asymptotic bias among all translation equivariant estimates of location. However, the median efficiency of $2/\pi$ at the Gaussian model may be unacceptably low in some applications. This motivates one to solve the following problem for the above $\varepsilon$-contamination family: Among all location M-estimates, find the one which minimizes the maximum asymptotic bias subject to a constraint on efficiency at the Gaussian model. This problem is the dual form analog of Hampel's optimality problem of minimizing the asymptotic variance at the nominal model (e.g., the Gaussian model) subject to a bound on the gross-error sensitivity. We solve the global problem completely for the case of a known scale parameter. The main conclusion is that Hampel's heuristic is essentially correct: The resulting M-estimate is based on a $\psi$ function which is amazingly close, but not exactly equal, to the Huber/Hampel optimal $\psi$. It turns out that one pays only a relatively small price in terms of increase in maximal bias for increasing efficiency from 64% to the range 90-95%. We also present a conjectured solution to the problem, based on heuristic arguments and numerical calculations, when the nuisance scale parameter is unknown.