Likelihood ratios for eigenvalues in spiked multivariate models
In 1964 Alan James gave a remarkable classification of many of the eigenvalue distribution
problems of multivariate statistics. We show how the classification readily adapts
to contemporary `spiked models' - high dimensional data with low rank structure. In particular
we look at asymptotic approximations to the likelihood ratios when the number
of variables grows proportionately with sample size or degrees of freedom. High dimensions
bring phase transition phenomena, with quite different limit behavior for small and
large spike strengths. James' framework allows us to develop these in a unified way across
problems such as signal detection, matrix denoising, regression and canonical correlations.
Joint work with Alexei Onatski and Prathapa Dharmawansa.
Hommel's method for false discovery proportions
We study the combination of closed testing with local tests based on the Simes inequality that underlies Hochberg's and Hommel's methods for familywise error control in multiple testing. We show this combination can also be used to make simultaneous confidence statements for the false discovery proportion of arbitrary sets of hypotheses. These confidence statements are at least as powerful as those arising trivially from Hommel's method, and can be much more powerful. Calculation time is linear in set size after an initial preparatory step that takes m log(m) time for m hypotheses. By their simultaneity, the coverage of these confidence statements is guaranteed even if sets of interest are chosen post hoc. With this method, the role of the user and the method can be reversed in multiple testing. Where normally the user chooses the error rate and the multiple testing method the rejected set, now the user can choose the rejected set and the multiple testing method calculates the error rate. This new method has some interesting connections to the Benjamini & Hochberg FDR method, which we explore.
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Minimax theory for a class of non-linear statistical inverse problems
We study minimax estimation rates for a class of non-linear statistical inverse problems motivated by various applications, including density estimation, binary regression and spectral density estimation. We derive matching upper and lower bounds for pointwise loss using function-dependent rates that capture spatial heterogeneity of the function. The upper bound is obtained via easy to implement plug-in estimators based on hard-wavelet thresholding that are shown to be fully adaptive, both spatially and over Holder smoothness classes.
This is joint work with Johannes Schmidt-Hieber.
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