1 Prliminaries 2 1.1 Basic Probability concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Stochastic Differential Equations (SDE’s) . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.1 Langevin equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.2 Ito integral and basics of Ito calculus . . . . . . . . . . . . . . ....

We present a novel method for controlling the k-familywise error rate (k-FWER) in the linear regression setting using the knockoffs framework first introduced by Barber and Candès. Our procedure, which we also refer to as knockoffs, can be applied with any design matrix with at least as many observations as variables, and does not require knowing the noise variance. Unlike other multiple testin...

Multiple comparison procedures that control a family-wise error rate or false discovery rate provide an achieved error rate as the adjusted p-value or q-value for each hypothesis tested. However, since achieved error rates are not understood as probabilities that the null hypotheses are true, empirical Bayes methods have been employed to estimate such posterior probabilities, called local false...

Multiple hypothesis testing is a significant problem in nearly all neuroimaging studies. In order to correct for this phenomena, we require a reliable estimate of the Family-Wise Error Rate (FWER). The well known Bonferroni correction method, while being simple to implement, is quite conservative, and can substantially under-power a study because it ignores dependencies between test statistics....

The idea of modifying, and potentially improving, classical multiple testing methods controlling the familywise error rate (FWER) via an estimate of the unknown number of true null hypotheses has been around for a long time without a formal answer to the question whether or not such adaptive methods ultimately maintain the strong control of FWER, until Finner and Gontscharuk (2009) and Guo (200...

When efficacy of a treatment is measured by co-primary endpoints, efficacy is claimed only if for each endpoint an individual statistical test is significant at level α. While such a strategy controls the family-wise type I error rate (FWER), it is often strictly conservative and allows for no inference if not all null hypotheses can be rejected. In this paper, we investigate fallback tests, wh...

We study generalized bootstrapped confidence regions for the mean of a random vector whose coordinates have an unknown dependence structure. The dimensionality of the vector can possibly be much larger than the number of observations and we focus on a non-asymptotic control of the confidence level. The random vector is supposed to be either Gaussian or to have a symmetric bounded distribution. ...

Consider the problem of testing s null hypotheses simultaneously. In order to deal with the multiplicity problem, the classical approach is to restrict attention to multiple testing procedures which control the familywise error rate (FWE). The closure method of Marcus et al. (1976) reduces the problem of constructing such procedures to one of constructing individual tests that control the usual...