Marc Raimondo - CMI

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Summary (for Part II) T-W approximation to null distribution of Roy’s largest root: � Conventional percentiles: generally accurate to < 10% relative error � [Rough p−value assessments: qualitatively o.k. over many orders of magnitude] � Similar O(N −2/3 ) approximations for Wishart and Gaussian. Recommend: Tracy-Widom approximation replace F − lower bound in default package printouts Luminy 12/08 – p.28
I. Setting Outline 1. Single & Double Wishart, examples II. Largest Eigenvalue: Limiting Law (H0) 1. Gaussian 2. Laguerre/Single W. 3. Jacobi/Double W. III. Largest Eigenvalue: Concentration 1. Single W. 2. Double W. Luminy 12/08 – p.29
Page 1 and 2: Marc Raimondo Luminy 12/08 - p.1
Page 3 and 4: The Annals of Statistics 2004, Vol.
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Page 7 and 8: Marc Raimondo Memorial Session: - N
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Page 11 and 12: Joint density of eigenvalues Single
Page 13 and 14: Symmetric Gaussian matrices A - sym
Page 15 and 16: f(x) 0.35 0.3 0.25 0.2 0.15 0.1 0.0
Page 17 and 18: I. Setting Outline 1. Single & Doub
Page 19 and 20: Second order accuracy For {real, co
Page 21 and 22: I. Setting Outline 1. Single & Doub
Page 23 and 24: The Tracy-Widom approximation Let W
Page 25 and 26: Accuracy of approximate α th perce
Page 27: θ α log 10 n 1 0.5 p = 4 at 90th
Page 31 and 32: ‘Small’ vs. ‘Large’ Deviati
Page 33 and 34: Rmax as singular value Use SVD: X =
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Page 37 and 38: Determinant representation: Correla
Page 39 and 40: After rescaling: Sτ(s,t) = 1 2 Uni
Page 41 and 42: Orthogonal Case - Convergence P { m
Page 43 and 44: Tail p−values: s = 2 variables F

Marc Raimondo - CMI

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