Inference for possibly misspecified generalized linear models with nonpolynomial-dimensional nuisance parameters

It is routine practice in statistical modelling to first select variables and then make inference for the selected model as in stepwise regression. Such inference is made upon the assumption that the selected model is true. However, without this assumption, one would not know the validity of the inference. Similar problems also exist in high-dimensional regression with regularization. To address these problems, we propose a dimension-reduced generalized likelihood ratio test for generalized linear models with nonpolynomial dimensionality, based on quasilikelihood estimation that allows for misspecification of the conditional variance. The test has nearly oracle performance when using the correct amount of shrinkage and has robust performance against the choice of regularization parameter across a large range. We further develop an adaptive data-driven dimension-reduced generalized likelihood ratio test and prove that, with probability going to one, it is an oracle generalized likelihood ratio test. However, in ultrahigh-dimensional models the penalized estimation may produce spuriously important variables that deteriorate the performance of the test. To tackle this problem, we introduce a cross-fitted dimension-reduced generalized likelihood ratio test, which is not only free of spurious effects, but robust against the choice of regularization parameter. We establish limiting distributions of the proposed tests. Their advantages are highlighted via theoretical and empirical comparisons to some competitive tests. An application to breast cancer data illustrates the use of our proposed methodology.