Since their introduction by Dwass (1957) and Barnard (1963), Monte Carlo tests have attracted considerable attention. The aim of this paper is to give a unified approach that covers the case of an arbitrary null distribution in order to study the statistical properties of Monte Carlo tests under the null hypothesis and under the alternative. For finite samples we obtain bounds for the power of the Monte Carlo test with the original test that allow determination of the required simulation effort. Furthermore the concept of asymptotic (resp. local asymptotic) relative Pitman efficiency (ARPE, resp. LARPE) is adapted to Monte Carlo tests for the study of their asymptotic behaviour. The normal limit case is investigated in more detail, leading to explicit formulas for ARPE and LARPE.
"Finite Sample Properties and Asymptotic Efficiency of Monte Carlo Tests." Ann. Statist. 14 (1) 336 - 347, March, 1986. https://doi.org/10.1214/aos/1176349860