Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 62F10, 62F12

Retrieve articles in all journals with MSC: 62F10, 62F12


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1984-0756045-0
PII: S 0002-9947(1984)0756045-0
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