content_image/305124.jpg content_image/305131.jpg content_image/305130.jpg
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 

<img src='content_image/305124.jpg'>

<img src='content_image/305131.jpg'>

<img src='content_image/305130.jpg'>

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 ) . 

