  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.
