On the Effects of Moderate Multivariate Nonnormality on Roy's Largest Root Test

Abstract

The asyniptotic distribution of Roy's largest root statistic in multivariate Edgeworth populations is expanded to terms of the first order. To this order, the effects of nonnormality are expressed by Mardia's measures of multivariate skewness and kurtosis. together with a supplementary skewness measure. Tables of corrections to the nominal significance level are presented for the I\-sample MANOVA situation. Comparison with simulation results indicates that the correction terms provide a useful indication both of the magnitude and the direction of the effect of nonnormality on the nominal 5 percent Type I error, even when the underlying population is not well represented by an Edgeworth expansion, provided that the skewness and kurtosis are not too large.KEY WORDS: Roy's largest root test: Multivariate skewness and kurtosis; MANOVA: Linear differential equations; Doubly noncentral beta distribution.

Statistics on Special Manifolds

“…Other results of invariant polynomials may be given in the text when they are needed. A more detailed discussion, including the proofs and applications in multivariate distribution theory, of invariant polynomials may be found in Davis (1979Davis ( , 1980Davis ( , 1981Davis ( , 1982, Chikuse (1980Chikuse ( , 1981Chikuse ( , 1985Chikuse ( , 1986aChikuse ( , 1992b, and Davis (1986a, 1986b); see Phillips (1980Phillips ( , 1984 and many others for applications to distribution theory in econometrics. We use the notation A[q+1,r) = (Aq+1' ... ,Ar) and A[q+ l,r] = (Aq+1' ... ,Ar).…”

Some properties of invariant polynomials with matrix arguments and their applications in econometrics

“…These polynomials have also been applied to multivariate distribution theory (see e.g. Chikuse [2], Davis [5], [7], [8], [9], Hayakawa [10], [11], Hillier et al [13], Mathai and Pillai [18], Phillips [19]). …”

Invariant Polynomials and Related Tests

“…Davis (1976) has expressed the joint p. d. f. of the latent roots of a non-central Wishart matrix in terms of Laguerre polynomials due to Constantine (1966). Davis (1980b) Similar arguments are applied for Roy's largest root test in Davis (1982a), the one sample Hotelling's T 2 test in Davis (1982b) and the problem of Roy and Gnanadesikan (1959)'s multivariate Model II with k-way classification in Davis (1982c).…”