## Monte Carlo tests based on pivotal statistics

In the following, we briefly outline the MC test methodology as it applies to the pivotal statistic context and a right-tailed test; for a more detailed discussion, see Dufour (1995) and Dufour and Kiviet (1998).

Let S0 denote the observed test statistic S, where S is the test criterion. We assume S has a unique continuous distribution under the null hypothesis (S is a continuous pivotal statistic). Suppose we can generate N iid replications, Sj, j = 1,…, N, of this test statistic under the null hypothesis. Compute

In other words, NGn(S0) is the number of simulated statistics which are greater or equal to S0, and provided none of the simulated values Sj, j = 1, …, N, is equal to S0, PN(S0) = N – NGn(S0) + 1 gives the rank of S0 among the variables S0, S1, . . ...

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