0.1 0.2 0.2 0.3 0.3 0.3 0.4 0.4 0.4 0.5 0.5 0.5 0.6 0.6 0.6 0.7 0.7 0.7 0.7 0.8 0.8 0.8 0.8 0.9 0.9 0.9 0.9 h$_{1}$/c 1/2 h 2 /c 1/2 Power envelope 0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 0.1 0.1 0.2 0.2 0.2 0.3 0.3 0.3 0.4 0.4 0.4 0.5 0.5 0.5 0.6 0.6 0.6 0.6 0.7 0.7 0.7 0.7 0.8 0.8 0.8 0.8 0.9 0.9 0.9 0.9 h$_{1}$/c 1/2 h 2 /c 1/2 Power of the LR test 0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 Figure 2: The asymptotic power envelope β$_{λ}$ ( h ) and the asymptotic power of the LR test based on λ ; r = 2 , asymptotic size is 0 . 05. Krichman and Nadler (2009), Nadakuditi and Silverstein (2010), Onatski (2009), Patterson et al (2006), Perry and Wolf (2010), and Tracy and Widom (2009)), have trivial asymptotic power (that is, the asymptotic power, which equals the asymptotic size) in the region h ∈ [0 $_{,}$√$_{c}$$_{)}$ $^{r}$. It is because the asymptotic behavior of any finite number of the largest sample covariance eigenvalues when h ∈ [0 $_{,}$√$_{c}$$_{)}$ r is not different from their behavior when the data are pure noise (P´ ech´ e, 2003). As was mentioned above, signal detection tests can be interpreted as tests of sphericity. Vice versa, previously proposed sphericity tests, can, in principle, be used for signal detection. In the Supplementary Appendix, we use Theorem 1 along with Le Cam’s third lemma to derive asymptotic powers of several such tests against “spiked covariance” alternatives. The derived asymptotic powers turn out to be much lower than the asymptotic power envelopes β$_{λ}$ ( h ) and β$_{µ}$ ( h ). However, we feel that this comparison is somewhat unfair to the sphericity tests because they are typically designed against general alternatives, as opposed to “the spiked covariance” alternatives. Therefore, and to save space, we do not report these results here. 20