On the likelihood ratio test for envelope models in multivariate linear regression

We investigate the likelihood ratio test for a hypothesis regarding the dimension of the Sigma-envelope of span(beta) in a multivariate linear regression model. The asymptotic null distribution of the likelihood ratio statistic is obtained as some nuisance parameters approach infinity. A saddlepoint approximation is also given for this limiting distribution. The accuracy of this approximation and its comparison to the standard chi-squared approximation are assessed via simulation. The results can be used in a similar test for partial envelope models