Large deviations and estimation in infinite-dimensional models (Q1115055)

Consider a random sample from a statistical model with an unknown, and possibly infinite-dimensional, parameter - e.g., a nonparametric or semiparametric model - and a real-valued functional T of this parameter which is to be estimated. The objective is to develop bounds on the (negative) exponential rate at which consistent estimates converge in probability to T, or, equivalently, lower bounds for the asymptotic effective standard deviation of such estimates - that is, to extend work of \textit{R. R. Bahadur} [see Ann. Math. Stat. 38, 303-324 (1967; Zbl 0201.521) and ``Some limit theorems in statistics.'' (1971; Zbl 0257.62015)] from parametric models to more general (semiparametric and nonparametric) models. The approach is to define a finite-dimensional submodel, determine Bahadur bounds for a finite-dimensional model, and then `sup' or `inf' the bounds with respect to ways of defining the submodels; this can be constructed as a `directional approach', the submodels being in a specified `direction' from a specific model. Extension is made to the estimation of vector-valued and infinite-dimensional functionals T, by expressing consistency in terms of a distance, or, alternatively, by treating classes of real functionals of T. Several examples are presented.