Asymptotic expansions for general statistical models. With the assist. of W. Wefelmeyer
type: Item
instance of: scholarly article
title: Asymptotic expansions for general statistical models. With the assist. of W. Wefelmeyer
zbMATH Open document ID: 0578.62003
published in: Lecture Notes in Statistics
publication date: 1985-01-01T00:00:00Z
review text: In his book ''Contributions to a general asymptotic statistical theory.'' Lect. Notes Stat. 13 (1982; Zbl 0512.62001), the author tried to describe the local structure of a general family \({\mathcal P}\) of probability measures by its tangent space and the local behavior of a functional \(\kappa: P\to R^ k\) by its gradient and obtained asymptotic envelope power functions for tests and asymptotic bounds for the concentration of estimators, when statistical procedures are based on independent, identically distributed observations. As a continuation, in this book the author considers these asymptotic investigations not by limit distributions but by Edgeworth expansions (adding one term of order \(n^{-1/2}\) to the limit distributions). To do this, the author introduces concepts of ''degenerate convergence condition of order (b,c) (for short: \(DCC_{b,c})''\) and \(''DCC_{b,c}\)-differentiable with derivative g at a rate \(o(t^ a)''\), and with the aid of ''canonical gradients'' he studies the second order envelope power functions for composite hypotheses, second order bounds for the concentration of confidence bounds and median unbiased estimators, etc. Then, using these results, he considers various general statistical procedures and gives bounds of order \(o(n^{-1/2})\) for the efficiency of them. Many examples and explanations of the new notions are given. It seems that many problems are still open.
Mathematics Subject Classification ID: 62A01
Mathematics Subject Classification ID: 62F05
Mathematics Subject Classification ID: 62F12
Mathematics Subject Classification ID: 62-02
zbMATH DE Number: 3925959
zbMATH Keywords: differentiability
zbMATH Keywords: Edgeworth expansions
zbMATH Keywords: degenerate convergence condition
zbMATH Keywords: canonical gradients
zbMATH Keywords: second order envelope power functions
zbMATH Keywords: composite hypotheses
zbMATH Keywords: second order bounds
zbMATH Keywords: concentration of confidence bounds
zbMATH Keywords: median unbiased estimators
zbMATH Keywords: efficiency
zbMATH Keywords: examples
author: Johann Pfanzagl
reviewed by: Ken-ichi Yoshihara
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3925959