0 0.5 1 0.16 0.18 0.2 h$_{2}$/h$_{1}$ Envelope power 25% 0 0.5 1 0.35 0.4 0.45 h$_{2}$/h$_{1}$ Envelope power 50% 0 0.5 1 0.64 0.66 0.68 0.7 h$_{2}$/h$_{1}$ Envelope power 75% 0 0.5 1 0.82 0.84 0.86 0.88 h$_{2}$/h$_{1}$ Envelope power 90% Figure 4: Power of λ -based (solid lines) and µ -based (dashed lines) LR tests plot- ted against h$_{2}$/h$_{1}$, where ( h$_{1}$, h$_{2}$ ) are such that the respective asymptotic power envelopes β$_{λ}$ ( h ) and β$_{µ}$ ( h ) equal 25, 50, 75 and 90%. the actual r does not coincide with the value the test statistic is based on? For example, to detect a single signal, one can, in principle, use LR tests of the null hypothesis against alternatives with r = 1 , r = 2 , etc. How does the asymptotic powers of such tests compare? Figure 5 reports the asymptotic powers of the λ - and µ -based LR tests designed to detect alternatives with r = 1 (solid line) and r = 2 (dashed line), under single-spiked alternatives. As in Figures 2 and 3, we use the parametrization θ = √ − ln (1 − h $^{2}$/c ) for the single-spiked alternative. It appears that the two asymptotic powers are very close to each other; interestingly, neither of them dominates the other. Using LR tests designed against alternatives with r > 1 seems to be beneficial for detecting a single-spiked alternative with relatively small θ (and h ). In the remaining part of this section, we consider examples of some of the tests that have been proposed previously in the literature, and, in Proposition 6, derive their asymptotic power functions. 17