Robust Fixed Size Confidence Procedures for a Restricted Parameter Space

Abstract

Robust fixed size confidence procedures are derived for the location parameter $\theta$ of a sample of $N$ i.i.d. observations of a scalar random variable $Z$ with CDF $F(z - \theta)$. Here, $\theta$ is restricted to a closed interval $\Omega$ and the uncertainty in $F$ is modeled by an uncertainty class $\mathscr{F}$. These robust confidence procedures are, in turn, based on the solution of a related robust minimax decision problem that is characterized by a zero-one loss function, the parameter space $\Omega$ and the uncertainty class $\mathscr{F}$. Sufficient conditions for the existence of robust minimax and robust median-minimax estimators are delineated. Sufficient conditions on $\mathscr{F}$ are obtained such that (i) both types of rules are minimax within the class of nonrandomized odd monotone procedures and (ii) subject to additional conditions, both types of rules are globally minimax admissible Bayes procedures. The paper concludes with an examination of the asymptotic behavior of the robust median-minimax estimators and their extensions to the robust $\alpha$-minimax rules, which are based on the $\alpha$-trimmed mean.