A General Approach to the Optimality of Minimum Distance Estimators

Let $\Theta$ be an open subset of a separable Hilbert space, and ${\xi _n}(\theta )$, $\theta \in \Theta$, a sequence of stochastic processes with values in a (different) Hilbert space $B$. This paper develops an asymptotic expansion and an asymptotic minimax result for "estimates" ${\hat \theta _n}$ defined by ${\inf _\theta }|{\xi _n}(\theta )| = |{\xi _n}({\hat \theta _n})|$, where $| \cdot |$ is the norm of $B$. The abstract results are applied to study optimality and asymptotic normality of procedures in a number of important practical problems, including simple regression, spectral function estimation, quantile function methods, min-chi-square methods, min-Hellinger methods, minimum distance methods based on $M$-functionals, and so forth. The results unify several studies in the literature, but most of the ${\text {LAM}}$ results are new. From the point of view of applications, the entire paper is a sustained essay concerning the problem of fitting data with a reasonable, but relatively simple, model that everyone knows cannot be exact.