Bootstrap Test Procedure for Variance Components in Nonlinear Mixed Effects Models in the Presence of Nuisance Parameters and a Singular Fisher Information Matrix

We examine the problem of variance components testing in general mixedeffects models using the likelihood ratio test. We account for the presence ofnuisance parameters, i.e. the fact that some untested variances might also beequal to zero. Two main issues arise in this context leading to a non regularsetting. First, under the null hypothesis the true parameter value lies on theboundary of the parameter space. Moreover, due to the presence of nuisanceparameters the exact location of these boundary points is not known, whichprevents from using classical asymptotic theory of maximum likelihoodestimation. Then, in the specific context of nonlinear mixed-effects models,the Fisher information matrix is singular at the true parameter value. Weaddress these two points by proposing a shrinked parametric bootstrapprocedure, which is straightforward to apply even for nonlinear models. We showthat the procedure is consistent, solving both the boundary and the singularityissues, and we provide a verifiable criterion for the applicability of ourtheoretical results. We show through a simulation study that, compared to theasymptotic approach, our procedure has a better small sample performance and ismore robust to the presence of nuisance parameters. A real data application isalso provided.