Theorem 5. Suppose that C1), C2) and C3) hold. Then, under H$_{0}$ and provided $_{U$_{n}$}$Γ 2 n → P 2 $_{0}$Γ 2 0 > 0 in probability, ˆ q ub 0 . 95 ≜ 6 . 2 · ($_{U}$$_{n}$Γ 2 $_{n}$) 1 / 2 > q$_{0}$$_{.}$$_{95}$ is a valid upper bound in the sense of (6) . The proof of the result follows immediately by noting that P ( X > t ) ≤ (1 + t $^{2}$) − 1 for any random variable X with mean zero and unit variance in view of the one-sided Chebyshev’s inequality. This illustrates concretely that we can obtain a consistent test that controls type I error. In practice, we recommend either using the result of 2 whenever possible or estimating the eigenvalues of the matrix in (5). We note that the condition σ 2 R > 0 holds in many but not all examples of interest. Fortunately, the plausibility of this assumption can be evaluated analytically. In Section 5, we show that this condition does not hold in Example 4 and provide a way forward despite this. We present two analyses of the asymptotic behavior of our test under a fixed alternative. The first relies on ˆ P$_{n}$ providing a good estimate of P$_{0}$ . Under this condition, we give an inter- pretable limit distribution that provides insight into the behavior of our estimator under the alternative. As we show, surprisingly, ˆ P$_{n}$ need not be close to P$_{0}$ to obtain an asymptotically consistent test, even if the resulting estimate of ψ$_{0}$ is nowhere near the truth. In the second analysis, we give more general conditions under which our test will be consistent if H$_{1}$ holds. 19 4.1 Consistency under a fixed alternative 4 Asymptotic behavior under the alternative