On confidence regions in canonical discriminant analysis
There are given q \(p\)-variate normal populations \(\Pi_ j\), \(j=1,\dots,q\) with means \(\mu_ j\) and the same covariance matrix \(\Sigma\), where \(\mu_ j\) and \(\Sigma\) are unknown, and there are \(N_ j\) observations \(x_{jk}\) from the \(j\)-th population \(\Pi_ j\) \((k=1,\dots,N_ j\), \(\Sigma N_ j=N)\). Let \(y_ j\) be the canonical discriminant variates based on the first \(s(<p)\) meaningful discriminant variates, and \(\bar y_ j=c'(\bar x_ j-\bar x)\), \(\eta_ j=c'(\mu_ j-\bar\mu)\), \(C=[c_ 1,\dots,c_ s]\) being the set of coefficient vectors. Confidence regions for \(\eta_ j\) corrected under the sampling variability of the canonical discriminant vectors are obtained as hyperspheres centered at \(\bar y_ j\) and having squared radii \(\bigl\{(1-N_ j/N)/N_ j\bigr\}\chi^ 2_{s,\alpha}\), where \(\chi^ 2_{s,\alpha}\) is the upper \(\alpha\) point of the \(\chi^ 2\)- distribution with \(s\) degrees of freedom --- instead of the radii \(\chi^ 2_{s,\alpha}/N_ j\) used up to now. As asymptotic confidence regions for the canonical discriminant values \(y_ j\) of a new observation \(x_ j\) from \(\Pi_ j\) hyperspheres centered at \(\bar y_ j\) and having squared radii \(\chi^ 2_{s,\alpha}\) have been used. More accurate confidence coefficients of the last confidence regions up to the order \(N^{-1}\) are given, and in the special case \(q=2\) \((s=1)\) there is \[ \chi^ 2_{1,\alpha}\to\chi^ 2_{1,\alpha}\bigl[1+n^{- 1}\bigl\{2p-1/2+N_ 2/N_ 1+(1/2)\chi^ 2_{1,\alpha}\bigr\}\bigr]+O(n^{-2}), \] where \(n=N-q\).
scientific article; zbMATH DE number 169013
scientific article