The Studentized maximum modulus (SMM) distribution is the
distribution of $$\max_{i=1,\ldots,m}\, |Z_i| \,/\, S,$$
where \(Z_1, \ldots, Z_m\) are independent standard normal variates
and \(S\) is independent of the \(Z_i\) and equal to
\(\sqrt{\chi^2_{df} / df}\).
What the SMM is used for. The SMM critical value is the
multiplier that turns a set of \(m\) individual estimates into a family
of simultaneous confidence intervals, or equivalently a family of tests,
that jointly control the family-wise error rate at level alpha_level.
Because the modulus is the largest of \(m\) standardized statistics in
absolute value, requiring that maximum to clear the critical value bounds
the chance of any one of the \(m\) intervals failing to cover (or any
one of the \(m\) tests producing a false positive). Maxwell, Delaney,
and Kelley (2027, Chapter 5) describe this use in the context of the
multiple-comparisons problem: when a researcher forms several means or
contrasts and wants the stated coverage to hold across the whole set
rather than one interval at a time, the SMM supplies the simultaneous
critical value. A pair-by-pair construction, applied to \(m\)
comparisons, is \(\hat\psi_i \pm c_{\alpha;m,df}\,\mathit{SE}_{\hat\psi_i}\)
with \(c_{\alpha;m,df}\) the SMM critical value returned here. When
\(m = 1\) it reduces to the ordinary two-sided t critical value.
How it is computed. The \(m\) statistics are independent given
the common scale estimate \(S\), so the joint distribution factorizes
after conditioning on \(S\) and the \(m\)-dimensional integral that
defines the maximum modulus collapses to a single one-dimensional integral,
$$P\!\left(\max_i |Z_i|/S \le c\right) =
\int_0^\infty \bigl[\,2\,\Phi(c\,s) - 1\,\bigr]^{m}\, f_S(s)\; ds,$$
where \(f_S\) is the density of \(S = \sqrt{\chi^2_{df}/df}\). This
package evaluates that integral with integrate and
inverts it with uniroot, so the returned value is
deterministic and accurate to the solver tolerance (there is no Monte
Carlo simulation and no random seed). When df = Inf the scale is
degenerate at 1 and the closed form
\(c = \Phi^{-1}\!\bigl((1 + (1-\alpha)^{1/m})/2\bigr)\) is returned; when
\(m = 1\) the value is exactly \(t_{1-\alpha/2,\,df}\).