Simultaneous likelihood-based bootstrap confidence sets for a large number of models

The paper studies a problem of constructing simultaneous likelihood-based confidence sets. We consider a simultaneous multiplier bootstrap procedure for estimating the quantiles of the joint distribution of the likelihood ratio statistics, and for adjusting the confidence level for multiplicity. Theoretical results state the bootstrap validity in the following setting: the sample size $n$ is fixed, the maximal parameter dimension $p_{\textrm{max}}$ and the number of considered parametric models $K$ are s.t. $(\log K)^{12}p_{\max}^{3}/n$ is small. We also consider the situation when the parametric models are misspecified. If the models' misspecification is significant, then the bootstrap critical values exceed the true ones and the simultaneous bootstrap confidence set becomes conservative. Numerical experiments for local constant and local quadratic regressions illustrate the theoretical results.