A general approach to the optimality of minimum distance estimators

Abstract

Let Θ \Theta be an open subset of a separable Hilbert space, and ξ n ( θ ) {\xi _n}(\theta ) , θ ∈ Θ \theta \in \Theta , a sequence of stochastic processes with values in a (different) Hilbert space B B . This paper develops an asymptotic expansion and an asymptotic minimax result for "estimates" θ ^ n {\hat \theta _n} defined by inf θ | ξ n ( θ ) | = | ξ n ( θ ^ n ) | {\inf _\theta }|{\xi _n}(\theta )| = |{\xi _n}({\hat \theta _n})| , where | ⋅ | | \cdot | is the norm of B 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 M -functionals, and so forth. The results unify several studies in the literature, but most of the LAM {\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.