A note on hypothesis testing in the envelope model

Abstract

Inference in response envelope model, which is developed in Cook, Li, and Chiaromonte (2010), is investigated. The accuracy of p-value from Wald test statistic for hypothesis testing under envelope model is of particular interest. The covariance matrix of the envelope model parameter estimator has been derived under the assumption that the envelope subspace dimension u is fixed. Estimating u might lead to an underestimation of the covariance matrix of the estimator of envelope model parameter. Our simulation study shows that such issue exists even for testing a simple one-dimensional hypothesis. Eck and Cook (2017) proposed a model average estimator, but the dimension reduction aspect from response envelope model would become uncertain. In this paper, we propose to use a modified test statistic that involves the covariance estimate of envelope estimator by residual bootstrapping method. The accuracy of p-value for this modified test statistic, the original Wald test statistic, and the test statistic based on model average estimator is compared, via simulation study.