Example 3 (The Bai et al. (2009) “corrected” LRT of H$_{0}$ : Σ = I .) When n > p, Bai et al. (2009) propose to use a corrected version of the likelihood ratio statistic to test the equality of the population covariance ma- trix to the identity matrix against general alternatives. Under the null, CLR d → N ( − 1 2 ln (1 − c ) , − 2 ln (1 − c ) − 2 c ) (still, as n, p → c ∞ ). The null hypothesis is rejected at asymptotic level α whenever CLR + 1 2 ln (1 − c ) is larger than ( − 2 ln (1 − c ) − 2 c ) 1 / $^{2}$Φ − $^{1}$(1 − α ) . Example 4 (Tracy-Widom-type tests of H$_{0}$ : Σ = I .) Let ϕ ( λ$_{1}$, ..., λ$_{r}$ ) be any function of the r largest eigenvalues increasing in all its arguments. The asymp- totic distribution of ϕ ( λ$_{1}$, ..., λ$_{r}$ ) under the null is determined by the functional form of ϕ ( · ) and the fact that where TW ( r ) denotes the r -dimensional Tracy-Widom law of the first kind, σ$_{n,c}$ = n 2 / $^{3}$c1 / $^{6}$(1 + √$_{c}$$_{)}$ − 4 / 3 and ν$_{c}$ = (1 + √$_{c}$$_{)}$ $^{2}$. Call Tracy-Widom-type tests all tests that reject the null whenever ϕ ( λ$_{1}$, ..., λ$_{r}$ ) is larger than the corresponding asymp- totic critical value obtained from (18). Example 5 (The Cai-Ma (2012) minimax test of H$_{0}$ : Σ = I .) Cai and Ma (2012) propose to use a U-statistic where ℓ ( X$_{1}$, X$_{2}$ ) = ( X ′ $_{1}$X$_{2}$ ) 2 − ( X ′ $_{1}$X$_{1}$ + X ′ $_{2}$X$_{2}$ ) + p, to test the hypothesis that the population covariance matrix is the unit matrix. Under the null, as n, p → c ∞ , T$_{n}$ d → N (0 , 4 c $^{2}$) . The null hypothesis is rejected at asymptotic level α whenever T$_{n}$ is larger than 2 √ p ( p + 1) /n ( n − 1)Φ − $^{1}$(1 − α ) . Cai and Ma (2012) show that this test is rate-optimal against general alternatives from a minimax point of view. 19 T$_{n}$ = 2 n ( n − 1) ∑ 1 ≤ i<j ≤ n ℓ ( X$_{i}$, X$_{j}$ ) , ( σ$_{n,c}$ ( λ$_{1}$ − ν$_{c}$ ) , ..., σ$_{n,c}$ ( λ$_{r}$ − ν$_{c}$ )) d → T W ( r ) , (18) CLR = tr ˆ Σ − ln det ˆ Σ − p − p ( 1 − ( 1 − n p ) ln ( 1 − p n )