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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A general approach to the optimality of minimum distance estimators

Author: P. W. Millar
Journal: Trans. Amer. Math. Soc. 286 (1984), 377-418
MSC: Primary 62F10; Secondary 62F12
MathSciNet review: 756045
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Abstract: 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 }\vert{\xi _n}(\theta )\vert = \vert{\xi _n}({\hat \theta _n})\vert$, where $ \vert \cdot \vert$ 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.

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Keywords: Local asymptotic minimax, minimum distance estimator, robust estimator, Cramer-von Mises statistic, minimum chi-square, differentiable statistical functional, abstract Wiener space
Article copyright: © Copyright 1984 American Mathematical Society

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