Corollary 1 Under assumptions of Lemma 1, An alternative way of deriving (11) is to apply l’Hˆ opital’s rule to the Harish- Chandra/Itzykson-Zuber determinantal formula the right-hand side of which is degenerate because A is rank-deficient. We include a proof of (11) that uses this approach in the Supplementary Appendix. The proof is elementary in the sense that it does not rely on properties of Jack polynomials. Combining Proposition 1 and Corollary 1 leads to useful contour integral repre- sentations of the likelihood ratios L ( h ; λ ) and L ( h ; µ ). We now introduce new notation to express such representations in a convenient form. For any z ∈ K , let us define a random variable where F $_{p}$( λ ) is the cumulative distribution function of the Marchenko-Pastur dis- tribution with a point mass of max ( 0 , 1 − c − 1 p ) at zero, where c$_{p}$ = p/n, and density Δ$_{p}$ ( z ) = p ∑ j =1 ln ( z − λ$_{j}$ ) − p ∫ ln ( z − λ ) d F $_{p}$( λ ) , (13) ψ$_{p}$ ( x ) = 1 2 πc$_{p}$x √ ( ¯ b$_{p}$ − x ) ( x − ¯ a$_{p}$ ) , (14) 4 Likelihood ratios as contour integrals $_{0}$F (1) 0 ( A, B ) = ( − 1) r ( r − 1) / 2 V$_{r}$ ( A ) r ∏ j =1 ( p − j )! a p r j det 1 ≤ i,j ≤ r ( 1 2 π i ∮ K e a$_{i}$$^{z}$zj − $^{1}$d z ∏ p s $_{=1}$( z − b$_{s}$ ) . (11) $_{0}$F (1) 0 ( A, B ) = ∏ p − 1 j =1 j ! V$_{p}$ ( A ) V$_{p}$ ( B ) det 1 ≤ i,j ≤ p ( e a$_{i}$$^{b$_{j}$}$) , (12) 13