A generalized projection pursuit procedure and its significance level
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 \(X^*\).
scientific article; zbMATH DE number 1117455
scientific article