Higher order asymptotic theory for discriminant analysis in exponential families of distributions

Masanobu Taniguchi*

*この研究の対応する著者

研究成果: Article査読

4 被引用数 (Scopus)

抄録

This paper deals with the problem of classifying a multivariate observation X into one of two populations Π1: p(x; w(1)) ∈ S and Π2: p(x; w(2)) ∈ S, where S is an exponential family of distributions and w(1) and w(2) are unknown parameters. Let I; be a class of appropriate estimators (ŵ(1), ŵ(2)) of (w(1), w(2) based on training samples. Then we develop the higher order asymptotic theory for a class of classification statistics D = [Ŵ | Ŵ = log(p(X; ŵ(1))/p(X; ŵ(2))), (ŵ(1), ŵ(2)) ∈ I;]. The associated probabilities of misclassification of both kinds M(ŵ) are evaluated up to second order of the reciprocal of the sample sizes. A classification statistic Ŵ is said to be second order asymptotically best in D if it minimizes M(Ŵ) up to second order. A sufficient condition for Ŵ to be second order asymptotically best in D is given. Our results are very general and give us a unified view in discriminant analysis. As special results, the Anderson W, the Cochran and Bliss classification statistic, and the quadratic classification statistic are shown to be second order asymptotically best in D in each suitable classification problem. Also, discriminant analysis in a curved exponential family is discussed.

本文言語English
ページ(範囲)169-187
ページ数19
ジャーナルJournal of Multivariate Analysis
48
2
DOI
出版ステータスPublished - 1994 2
外部発表はい

ASJC Scopus subject areas

  • 統計学および確率
  • 数値解析
  • 統計学、確率および不確実性

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