  In applied settings, tests of hypothesis where a nuisance parameter is only
identifiable under the alternative often reduces into one of Testing One
Hypothesis Multiple times (TOHM). Specifically, a fine discretization of the
space of the non-identifiable parameter is specified, and the null hypothesis
is tested against a set of sub-alternative hypothesis, one for each point of
the discretization. The resulting sub-test statistics are then combined to
obtain a global p-value. In this paper, we discuss a computationally efficient
inferential tool to perform TOHM under stringent significance requirements,
such as those typically required in the physical sciences, (e.g., p-value
$<10^{-7}$). The resulting procedure leads to a generalized approach to perform
inference under non-standard conditions, including non-nested models
comparisons.
