content_image/300887.jpg content_image/300888.jpg content_image/300884.jpg content_image/300885.jpg
Corollary 1 Under assumptions of Lemma 1, 

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An alternative way of deriving (11) is to apply l’Hˆ opital’s rule to the Harish- Chandra/Itzykson-Zuber determinantal formula 

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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. 

## 4 Likelihood ratios as contour integrals 

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 

<img src='content_image/300884.jpg'>

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 

<img src='content_image/300885.jpg'>

