Parametric Robustness: Small Biases can be Worthwhile

Abstract

We study estimation of the parameters of a Gaussian linear model $\mathscr{M}_0$ when we entertain the possibility that $\mathscr{M}_0$ is invalid and a larger model $\mathscr{M}_1$ should be assumed. Estimates are robust if their maximum risk over $\mathscr{M}_1$ is finite and the most robust estimate is the least squares estimate under $\mathscr{M}_1$. We apply notions of Hodges and Lehmann (1952) and Efron and Morris (1971) to obtain (biased) estimates which do well under $\mathscr{M}_0$ at a small price in robustness. Extensions to confidence intervals, simultaneous estimation of several parameters and large sample approximations applying to nested parametric models are also discussed.