Frequentist-Bayesian Monte Carlo test for mean vectors in high dimension

Conventional methods for testing the mean vector of a P-variate Gaussian distribution require a sample size N greater than or equal to P. But, in high dimensional situations, that is when N is smaller than P, special and new adjustments are needed. Although Bayesian-empirical methods are well-succeeded for testing in high dimension, their performances are strongly dependent on the actual unknown covariance matrix of the Gaussian random vector. In this paper, we introduce a hybrid frequentist-Bayesian Monte Carlo test and prove that: (i) under the null hypothesis, the performance of the proposed test is invariant with respect to the real unknown covariance matrix, and (ii) the decision rule is valid, which means that, in terms of expected loss, the performance of the proposed procedure can always be made as good as the exact Bayesian test and, in terms of type I error probability, the method is always of α level for arbitrary α∈(0,1).