  Inference about a scalar parameter of interest typically relies on the
asymptotic normality of common likelihood pivots, such as the signed likelihood
root, the score and Wald statistics. Nevertheless, the resulting inferential
procedures are known to perform poorly when the dimension of the nuisance
parameter is large relative to the sample size and when the information about
the parameters is limited. In many such cases, the use of asymptotic normality
of analytical modifications of the signed likelihood root is known to recover
inferential performance. It is proved here that parametric bootstrap of
standard likelihood pivots results in as accurate inferences as analytical
modifications of the signed likelihood root do in stratified models with
stratum specific nuisance parameters. We focus on the challenging case where
the number of strata increases as fast or faster than the stratum samples size.
It is also shown that this equivalence holds regardless of whether constrained
or unconstrained bootstrap is used. This is in contrast to when the number of
strata is fixed or increases slower than the stratum sample size, where we show
that constrained bootstrap corrects inference to a higher order than
unconstrained bootstrap. Simulation experiments support the theoretical
findings and demonstrate the excellent performance of bootstrap in extreme
scenarios.
