0 0.5 1 0 0.5 1 0 0.5 1 h$_{1}$/c 1/2 h$_{2}$/c 1/2 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.6 0.7 0.7 0.7 0.8 0.8 0.8 0.9 0.9 0.9 h$_{1}$/c 1/2 h 2 /c 1/2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 0 0.5 1 0 0.5 1 0 0.5 1 h$_{1}$/c 1/2 h$_{2}$/c 1/2 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.5 0.6 0.6 0.6 0.7 0.7 0.7 0.8 0.8 0.8 0.9 0.9 0.9 h$_{1}$/c 1/2 h 2 /c 1/2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 Figure 1: The power envelopes β$_{λ}$ ( h ) (upper panel) and β$_{µ}$ ( h ) (lower panel) for α = 0 . 05 , as functions of $_{h/}$√$_{c}$ = ( h$_{1}$, h$_{2}$ ) $_{/}$√$_{c}$$_{.}$ Proposition 4 Let Φ denote the standard normal distribution function. Then, Figure 1 shows the asymptotic power envelopes β$_{λ}$ ( h ) and β$_{µ}$ ( h ) as functions of h$_{1}$$_{/}$√$_{c}$ and h$_{2}$$_{/}$√$_{c}$ when h = ( h$_{1}$, h$_{2}$ ) is two-dimensional. It is important to realize that the asymptotic power envelopes derived in Propo- sition 4 are valid not only for λ - and µ -based tests but also for any test invariant under left orthogonal transformations of the observations ( X ↦→ QX, where Q is a p × p orthogonal matrix), and for any test invariant under multiplication by any non-zero constant and left orthogonal transformations of the observations ( X ↦→ aQX, where a ∈ R + 0 and Q is a p × p orthogonal matrix), respectively. 13 β$_{λ}$ ( h ) = 1 − Φ   $_{Φ}$− $^{1}$(1 − α ) − √ − 1 2 r ∑ i,j =1 ln ( 1 − h$_{i}$h$_{j}$ c )   and (14) β$_{µ}$ ( h ) = 1 − Φ   $_{Φ}$− $^{1}$(1 − α ) − √ − 1 2 r ∑ i,j =1 ( ln ( 1 − h$_{i}$h$_{j}$ c ) + h$_{i}$h$_{j}$ c )   . (15)