A generalized projection pursuit procedure and its significance level
type: Item
instance of: scholarly article
title: A generalized projection pursuit procedure and its significance level
zbMATH Open document ID: 0923.62067
author: Kanta Naito
published in: Hiroshima Mathematical Journal
publication date: 1999-10-31T00:00:00Z
review text: In \textit{M. C. Jones} and \textit{R. Sibson}, J. R. Stat. Soc., Ser. A 150, 1-36 (1987; Zbl 0632.62059), a projection pursuit procedure which maximizes nonnormality was considered. In the paper under review the procedure is generalized to maximize nonellipticity. For this purpose for a \(p\)-dimensional r.v. \(X\) with mean \(\mu\) and covariance matrix \(\Sigma\) and for i.i.d. observations on \(X\) a generalized moment index (GMI) is defined as: \[ GMI(\alpha)= \sum^4_{k=3}[N^{-1}\sum^N_{j=1}\{H_k\bigl(\alpha' S^{-1/2}(X_j-\overline X)\bigr)-\theta_k\}/\sigma_k]^2. \] Here \(\alpha\) is a \(p\)-dimensional unit vector, \(H_k\) is the \(k\)-th order Hermite polynomial, \(\overline X\) and \(S\) are the sample mean vector and the sample covariance matrix, resp., \(\theta_k= E[H_k(\alpha' \sum^{-1/2} (X-\mu))]\), and \(\sigma^2_k\) is the asymptotic variance of \(H_k(\alpha' S^{-1/2}(X_1- \overline X))\) which is taken under the ellipticity hypothesis. This index simultaneously evaluates nonellipticity via the weighted sum of squares of skewness and kurtosis. An asymptotic behavior of GMI under ellipticity structure is explored and an approximation formula as \(N\to\infty\) of the significance level is derived. Several examples and numerical results are presented. Performances of GMI when a specified elliptical structure \(X\) is the least interesting are asymptotically evaluated under another elliptical structure \(X^*\).
Mathematics Subject Classification ID: 62H99
zbMATH DE Number: 1117455
zbMATH Keywords: exploratory projection pursuit
zbMATH Keywords: elliptically symmetric distributions
zbMATH Keywords: projection index
zbMATH Keywords: significance level
reviewed by: A. Kukush
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.32917/hmj/1206126967
OpenAlex ID: W1485539683
Wikidata QID: Q128899574
DOI: 10.32917/HMJ/1206126967
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description: scientific article; zbMATH DE number 1117455