0 2 4 6 0 0.2 0.4 0.6 0.8 1 h$_{1}$/c$^{1/2}$=0.3006 $_{2}$=(−log(1−h 2 $^{2}$/c))1/2 0 2 4 6 0 0.2 0.4 0.6 0.8 1 h$_{1}$/c$^{1/2}$=0.6683 $_{2}$=(−log(1−h 2 $^{2}$/c))1/2 0 2 4 6 0 0.2 0.4 0.6 0.8 1 h$_{1}$/c$^{1/2}$=0.9520 $_{2}$=(−log(1−h 2 $^{2}$/c))1/2 0 2 4 6 0 0.2 0.4 0.6 0.8 1 h$_{1}$/c$^{1/2}$=0.9995 $_{2}$=(−log(1−h 2 $^{2}$/c))1/2 Figure 2: Profiles of the asymptotic power of the λ -based LR test (solid lines) relative to the asymptotic power envelope (dotted lines) for different values of h$_{1}$$_{/}$√$_{c}$ under the alternative; α = 0 . 05 . readability, we use a different parametrization: θ$_{j}$ = √ − ln ( 1 − h 2 $_{j}$/c ) , i = 1 , ..., r . As h$_{j}$ varies in the region of contiguity [0 $_{,}$√$_{c}$$_{)}$$_{, θ}$$_{j}$ spans the entire half-line [0 , ∞ ) . Note that the asymptotic mean and autocovariance functions of the log likelihood ratios derived in Theorem 3 depend on h$_{j}$ only through h$_{j}$$_{/}$√$_{c}$ = √ 1 − $_{e}$− θ 2 j . Therefore, under the new parametrization, they depend only on θ = ( θ$_{1}$, ..., θ$_{r}$ ). The parameter θ plays the classical role of a “local parameter” in our setting. Figure 4 further explores the relationship between the asymptotic powers of the λ - and µ -based LR test and the corresponding asymptotic power envelopes when r = 2. We pick all values of h = ( h$_{1}$, h$_{2}$ ) satisfying inequality h$_{1}$ ≥ h$_{2}$ and such that the asymptotic power envelope for λ -based tests is exactly 25, 50, 75, and 90%. Then, we compute and plot the corresponding power of the λ -based LR test (solid lines) against h$_{2}$/h$_{1}$. The dashed lines show similar graphs for the 15