Identity-corrected random-permutation Monte Carlo p-value

Input: a measurable finite-group setup Measurable finite-group permutation setups, an observed $x$, an integer $B\ge 0$, and one of the two simulation designs below. Simulated transformations are independent of $X$.

1. With replacement. Draw $g_1,\dots ,g_B$ independently and uniformly from $G$. Set $g_0=e$, the identity, and return ${\widehat{p}}_{B,\mathrm{w}\mathrm{r}}(x):=\frac{1+{\sum }_{b=1}^B\mathbf{1}\{T(g_b\cdot x)\ge T(x)\}}{B+1}\mathrm{.}$ A simulated draw may itself equal $e$; it is still counted in addition to the distinguished identity.
2. Without replacement. Require $B\le \mathrm{\mid }G\mathrm{\mid }-1$. Draw a subset $S$ uniformly from the $B$-element subsets of $G\setminus \{e\}$, and return ${\widehat{p}}_{B,\mathrm{w}\mathrm{o}\mathrm{r}}(x):=\frac{\mathrm{\#}\{g\in \{e\}\cup S:T(g\cdot x)\ge T(x)\}}{B+1}\mathrm{.}$ This is a uniform-subset design; the selected transformations are distinct and the identity is adjoined.

In either design, a cutoff rule rejects when the returned value is at most $\alpha \in [0,1]$. For $B=0$ the only transformation is the distinguished identity and the returned value is $1$. Validity under an invariant null law, including ties, is a separate theorem on this page.