L1-norm error bounds for asymptotic expansions of multivariate scale mixtures and their applications to Hotelling's generalized T02

AbstractThis paper is concerned with the distribution of a multivariate scale mixture variate X=(X1,…,Xp)′ with Xi=SiZi, where Z1,…,Zp are i.i.d. random variables, Si>0(i=1,…,p), and {S1,…,Sp} is independent of {Z1,…,Zp}. First we obtain L1-norm error bounds for an asymptotic expansion of the density function of X in the multivariate case as well as in the univariate case. Then the results are applied in obtaining error bounds for asymptotic expansions of the null distribution of Hotelling's generalized T02-statistic. The special features of our results are that our error bounds are given in explicit and computable forms. Further, their orders are the same as ones of the usual order estimates, and hence the paper provides a new proof for validity of the asymptotic expansions

Non-uniform error bounds for asymptotic expansions of scale mixtures of distributions

Let X = [sigma]Z be the scale mixture of Z with the scale factor [sigma] > 0. We consider two type expansions G[delta],k(x) and [Phi][delta],k(x) as the approximations to the distribution function F(x) of X. In this paper we derive non-uniform error bounds in approximating F(x) by the asymptotic expansions G[delta],k(x) and [Phi][delta],k(x). The non-uniform bounds are improvements on the uniform bounds in the tail part of the distribution. The results are applied to the asymptotic expansions of t- and F-distributions.non-uniform error bounds asymptotic expansion distribution function scale mixture t- and F-distributions

Asymptotic expansions for the distributions of some functions of the latent roots of matrices in three situations

In this paper we derive asymptotic expansions for the distributions of some functions of the latent roots of the matrices in three situations in multivariate normal theory, i.e., (i) principal component analysis, (ii) MANOVA model and (iii) canonical correlation analysis. These expansions are obtained by using a perturbation method. Confidence intervals for the functions of the corresponding population roots are also obtained.Asymptotic distribution function of latent roots asymptotic expansions principal component analysis MANOVA model canonical correlation analysis perturbation method

The growth curve model with an autoregressive covariance structure

Growth curve model, autoregressive covariance structure, MLE's, asymptotic distributions, likelihood ratio statistic,