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High-Dimensional General Linear Hypothesis Tests via Spectral Shrinkage

Date:
Location:
MDS 220 & https://uky.zoom.us/j/84536644014?pwd=VHhvc1V0U1JWczF6cW1LNTZqMDRxdz09
Speaker(s) / Presenter(s):
Dr. Haoran Li - Columbia University

Abstract: In statistics, one of the fundamental inferential problems is to test a general linear hypothesis of regression coefficients under a linear model. The framework includes many well-studied problems such as two-sample tests for equality of population means, MANOVA, and others as special cases. The testing problem is well-studied in the classical multivariate analysis literature but remains underexplored under high-dimensional settings. Various classical invariant tests, despite their popularity in multivariate analysis, involve the inverse of the residual covariance matrix, which is inconsistent or even singular when the dimension is at least comparable to the degree of freedom. Consequently, classical tests perform poorly and power enhanced procedures are in need. 

 

In this talk, I seek to regularize the spectrum of the residual covariance matrix by flexible shrinkage functions. A family of rotation-invariant tests is proposed. For illustration purposes, we focus on ridge-type regularization in this talk.  The asymptotic normality of the test statistics under the null hypothesis is derived in the setting where dimensionality is comparable to the sample size. The asymptotic power of the proposed test is studied under a class of local alternatives. 

 

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