MCTests

MCTests, short for Monte Carlo tests, refer to a family of statistical hypothesis tests in which the reference distribution of a test statistic is approximated by Monte Carlo simulation rather than derived analytically. They are especially useful when the exact distribution under the null hypothesis is intractable or unknown, or when data or model complexity makes traditional theory unreliable.

The general procedure involves specifying a null hypothesis and a data-generating process consistent with that null. Synthetic data are generated by resampling, permutations, or by simulating from a fitted null model. For each simulated dataset, the test statistic is computed. The collection of simulated statistics forms an empirical null distribution, and the p-value is the proportion of simulations whose statistic is as extreme as or more extreme than the observed value. Permutation tests are a common special case of Monte Carlo tests when exchangeability holds, while bootstrap-based Monte Carlo tests use resampling with replacement.

MCTests are applied in a variety of settings, including goodness-of-fit tests, tests of independence, regression diagnostics, and complex or high-dimensional models where classical asymptotic results may be unreliable. They are also useful when the sampling distribution depends on nuisance parameters that are difficult to estimate analytically.

Practical considerations include the number of simulations, which controls Monte Carlo error, and computational cost. Reproducibility is achieved by fixing random seeds. When multiple tests are conducted, p-values may require adjustment. Extensions include importance sampling and sequential Monte Carlo to improve efficiency and accuracy. Software implementations exist in many statistical packages for R and Python.