Optimal generalized truncated sequential Monte Carlo test.

Abstract

When it is not possible to obtain the analytical null distribution of a test statistic U, Monte Carlo hypothesis tests can be used to perform the test. Monte Carlo tests are commonly used in a wide variety of applications, including spatial statistics, and biostatistics. Conventional Monte Carlo tests require the simulation of m independent copies from U under the null hypothesis, what is computationally intensive for large data sets. Truncated sequential Monte Carlo designs can be performed to reduce computational effort in such situations. Different truncated sequential procedures have been proposed. They work under restrictive assumptions on the distribution of U aiming to bound the power loss and to reduce execution time. Since the use of Monte Carlo tests are based on the situations where the null distribution of U is unknown, their results are not valid for the general case of any test statistic. In this paper, we derive an optimal scheme for truncated sequential Monte Carlo hypothesis tests. This scheme minimizes the expected number of simulations under any alternative hypothesis, and bounds the power loss in arbitrarily small values. The first advantage from this scheme is that the results concerning the power and the expected time are valid for any test statistic. Also, we present practical examples of optimal procedures for which the expected number of simulations are reduced by 60% in comparison with some of the best procedures in the literature.