0 2 4 6 0.2 0.4 0.6 0.8 1 =(−log(1−h $^{2}$/c)) 1/2 −based LR test 0 2 4 6 0.2 0.4 0.6 0.8 1 =(−log(1−h $^{2}$/c)) 1/2 −based LR test Figure 5: Asymptotic power of the λ -based (left panel) and µ -based (right panel) LR tests. Solid line: power when r = 1 is correctly assumed. Dashed line: power when r = 2 is incorrectly assumed. Example 1 (John’s (1971) test of sphericity H$_{0}$ : Σ = σ $^{2}$I. ) John (1971) pro- poses testing the sphericity hypothesis θ = 0 against general alternatives based on the test statistic where ˆ Σ is the sample covariance matrix. He shows that, when n > p, such a test is locally most powerful invariant. Ledoit and Wolf (2002) study John’s test when n, p → c ∞ . They prove that, under the null, nU − p d → N (1 , 4) . Hence, the test with asymptotic size α rejects the null of sphericity whenever 1 2 ( nU − p − 1) > Φ − $^{1}$(1 − α ) . Example 2 (The Ledoit-Wolf (2002) test of H$_{0}$ : Σ = I. ) Ledoit and Wolf (2002) propose as a test statistic for testing the hypothesis that the population covariance ma- trix is the unit matrix. They show that, under the null, nW − p d → N (1 , 4) . As in the previous example, the null is rejected at asymptotic size α whenever 1 2 ( nW − p − 1) > Φ − $^{1}$(1 − α ) . W = 1 p tr [ ( ˆ Σ − I ) 2 ] − p n [ 1 p tr ˆ Σ ] 2 + p n (17) U = 1 p tr     ˆ Σ (1 /p ) tr ( ˆ Σ ) − I$_{p}$  $^{2}$   , (16) 18