TablesTABLE 1.Coefficients of monomial symmetric functions M(Y)238in thezonal polynomial C(Y)Generalized binomial coficients ()268TABLE 2.Upper 5percentage points of -2log A*where A*is the modifiedTABLE3.likelihood ratio statistic for testing equality of r covariance310matrices (equal samplesizes)TABLE4.Lower 5and I percentage points of the ellipticity statisticV345for testing sphericity (=入I)TABLE S.Upper5andIpercentagepointsof-2logA*,whereA*is the modified likelihood ratio statistic360for testing that a covariance matrix equals a specified matrixTABLE6.Upper5andI percentagepointsof-2logA,whereAisthe likelihood ratio statistic for testing specified371values for the mean vector and covariance matrixTABLE7.Upper5percentagepointsof-2logA*,whereA*is the modified likelihood ratio statistic for testing514equalityof p normal populations (equal sample sizes)TABLE8.x adjustments to the likelihood ratio statistic for testing537independence: factor C for upper percentiles of-2plog A.TABLE9.x adjustments to Wilks likelihood ratio statistic W:595factorCforupperpercentilesof-NlogWxvii
Tables TABLE I. Coefficients of monomial symmetric functions MA( Y) in the zonal polynomial C,( Y) TABLE 2. Generalized binomial coefficients ( ) TABLE 3. Upper 5 percentage points of -210g A*, where A* is the modified likelihood ratio statistic for testing equality of r covariance matrices (equal sample sizes) TABLE 4. Lower 5 and I percentage points of the ellipticity statistic V TABLE 5. Upper 5 and I percentage points of -21og A*, where A' is the modified likelihood ratio statistic for testing that a covariance matrix equals a specified matrix TABLE 6. Upper 5 and I percentage points of -210g A, where A is the likelihood ratio statistic for testing specified values for the mean vector and covariance matrix TABLE 7. Upper 5 percentage points of -210g A*, where A* is the modified likelihood ratio statistic for testing equality of p normal populations (equal sample sizes) TABLE 8. x2 adjustments to the likelihood ratio statistic for testing TABLE 9. x2 adjustments to Wilks likelihood ratio statistic W: 238 268 31 0 for testing sphericity (2 = XI) 345 360 37 I 514 537 5 9s independence: factor C for upper percentiles of - 2plog A. factor C for upper percentiles of - Nlog W xvii
Commonly Used NotationRmEuclidean space of dimension m consisting of mX realvectorsdetdeterminantSmunit sphere in R" centeredat the originΛexterior or wedge productRereal parttrtraceexptretrA>0A is positivedefiniteVm.nStiefel manifold of n X m matrices with orthonormal columns0(m)GroupoforthogonalmXmmatricesdirect orKronecker product≥ x?IXIInormof xi.e.Sset of all m X m positive definite matricesgeneral lineargroup of m X m nonsingular real matricesSe(m,R)ee(m,R)affinegroup((B,c);BESe(m,R);cERm)C.(-)zonal polynomialM,(·)monomial symmetricfunction(α)xgeneralized hypergeometric coefficientIIXImax of absolute values of latent roots of the matrix X(6)generalized binomial coefficientL(.)generalized Laguerre polynomialm-variate normal distribution with mean μ and covarianceNm(μ,E)matrixZWm(n,2)m Xm matrix variate Wishart distribution with n degrees offreedom and covariance matrixZBetam(a,β)m X m matrix variate beta distribution with parameters α,βxix
Commonly Used Notation R det A Re tr etr A>O Vm, it O(m) c3 IlXll sm .S, GP(m, R) Euclidean space of dimension nz consisting of m X 1 real vectors determinant unit sphere in R" centered at the origin exterior or wedge product real part trace exp tr A is positive definite Stiefel manifold of n X m matrices with orthonormal columns Group of orthogonal m X m matrices direct or Kronecker product set of all m X m positive definite matrices general linear group of nr X m nonsingular real matrices affine group ((B,c); B~4t(m, R);cERm) zonal polynomial monomial symmetric function generalized hypergeometric coefficient max of absolute values of latent roots of the matrix X generalized binomial coefficient generalized Laguerre polynomial m-variate normal distribution with mean p and covariance matrix X m X m matrix variate Wishart distribution with n degrees of freedom and covariance matrix 2 m X m matrix variate beta distribution with parameters a, /3 xix
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