To “ a Practical Two - Step Method for Testing Moment Inequalities

for some prespecified value of α ∈ (0 1) rather than (3). In Section S.1.1 below, we first establish an upper bound on the power function of any test of (S.1) that satisfies (S.3) by deriving the most powerful test against any fixed alternative. We then describe our two-step procedure for testing (S.1) in Section S.1.2. Proofs of all results can be found in the Supplement Appendix. Before proceeding, note that, by sufficiency, we may assume without loss of generality that n = 1. Hence, the data consist of a single random variable W distributed according to the multivariate Gaussian distribution with unknown mean vector μ ∈Rk and known covariance matrix Σ. For 1 ≤ j ≤ k, we denote by Wj the jth component of W and by μj the jth component of μ. Note further that, because Σ is assumed known, we may assume without loss of generality that its diagonal consists of ones; otherwise, we can simply replace Wj by Wj divided by its standard deviation.