alternative equals five, and the signals have equal but rather weak strengths h$_{1}$ = ... = h$_{5}$ = 0 . 5. Then the best possible λ -based procedure for detecting such signals with the asymptotic probability of false detection fixed at 0.05 has asymptotic probability of correct detection 1 − Φ [ Φ − $^{1}$(0 . 95) − √ − 25 ln (1 − 0 . 25) ] ≈ 0 . 85. Unfortunately, constructing testing procedures with uniformly optimal power is hard because the log-likelihood process established in Theorem 1 is not of the Gaussian shift type, so that the statistical experiments we study are not locally asymptotically normal (LAN) ones. For the case of real-valued data and r = 1 , Onatski et al (2012) use numerical simulations to show that the asymptotic powers of the likelihood ratio (LR) tests based on λ and on µ are close to the respective asymptotic power envelopes β$_{λ}$ ( h ) and β$_{µ}$ ( h ). The λ - and µ -based LR tests of h = 0 against the alternative h ∈ ( 0 , ¯ h ) r reject the null if and only if , respectively, 2 sup$_{h}$$_{∈}$$_{(0}$$_{,}$ ¯ h $_{)}$ln L ( h ; λ ) and 2 sup$_{h}$$_{∈}$$_{(0}$$_{,}$ ¯ h $_{)}$ln L ( h ; µ ) are sufficiently large. As r grows, it becomes increasingly difficult to find the asymptotic critical values for the LR tests by simulation. This requires simulating an r -dimensional Gaussian random field with the covariance function and the mean function described in Theorem 1, which, for relatively large r , is computationally expensive. For r = 2 , Figure 2 shows the contour plots of the power envelope β$_{λ}$ ( h ) (left panel) and of the asymptotic power of the likelihood ratio test based on λ . We chose parameter ¯ h so that it is very close to the threshold √$_{c}$$_{, precisely}$ ¯ h = √ c (1 − e − $^{36}$) . We see that the contours of β$_{λ}$ ( h ) and of the asymptotic power of the λ -based LR test corresponding to the same value of these functions are relatively close to each other, which suggests that the LR test has good asymptotic power properties. More detailed analysis of the asymptotic and finite sample power of the LR test is, however, beyond the scope of this paper, and is left for future research. In contrast to the LR test, the popular signal detection procedures based on the information in a few of the largest eigenvalues of XX $^{′}$/n (see, for example, 19