Multiple hypothesis testing is concerned with maintaining low the
number of false positives when testing several hypotheses
simultaneously, while achieving a number of false negatives as
small as possible. Procedures should be distribution free and
robust with respect to known or possibly unknown dependence. This
thesis is related to modern approaches. After a review of the most
recent developments, we prove robustness of certain procedures
under weak dependence. We then propose a new class of procedures
and estimators for the proportion of false null hypotheses, i.e.,
the strength of the simoultaneous signal. In order to develop our
findings, we provide probability inequalities and related tools
under dependence. We show a pletora of applications. Main
motivating applications are in the field of genomics, but we also
show an innovating application to signal and image reconstruction
through wavelet thresholding.
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