Testing against a high-dimensional alternative in the generalized linear model: asymptotic type I error control

Testing a low-dimensional null hypothesis against a high-dimensional alternative in a generalized linear model may lead to a test statistic that is a quadratic form in the residuals under the null model. Using asymptotic arguments, we show that the distribution of such a test statistic can be approximated by a ratio of quadratic forms in normal variables, for which algorithms are readily available. For generalized linear models, the asymptotic distribution shows good control of type I error for moderate to small samples, even when the number of covariates in the model far exceeds the sample size.