Test statistics are often strongly dependent in large-scale multiple testing
applications. Most corrections for multiplicity are unduly conservative for
correlated test statistics, resulting in a loss of power to detect true
positives. We show that the Westfall--Young permutation method has
asymptotically optimal power for a broad class of testing problems with a
block-dependence and sparsity structure among the tests, when the number of
tests tends to infinity.

The ultimate goal of regression analysis is to obtain information about the
conditional distribution of a response given a set of explanatory variables.
This goal is, however, seldom achieved because most established regression
models only estimate the conditional mean as a function of the explanatory
variables and assume that higher moments are not affected by the regressors.
The underlying reason for such a restriction is the assumption of additivity of
signal and noise. We propose to relax this common assumption in the framework
of transformation models.

A conditional independence graph is a concise representation of pairwise
conditional independence among many variables. We propose Graphical Random
Forests (GRaFo) for estimating pairwise conditional independence relationships
among mixed-type, i.e. continuous and discrete, variables. The number of edges
is a tuning parameter in any graphical model estimator and there is no obvious
number that constitutes a good choice. Stability Selection helps choosing this
parameter with respect to a bound on the expected number of false positives
(error control).

In 1994, I came to Berkeley and was fortunate to stay there three years,
first as a postdoctoral researcher and then as Neyman Visiting Assistant
Professor. For me, this period was a unique opportunity to see other aspects
and learn many more things about statistics: the Department of Statistics at
Berkeley was much bigger and hence broader than my home at ETH Z\"urich and I
enjoyed very much that the science was perhaps a bit more speculative.

We propose an $\ell_1$-penalized estimation procedure for high-dimensional
linear mixed-effects models. The models are useful whenever there is a grouping
structure among high-dimensional observations, i.e. for clustered data. We
prove a consistency and an oracle optimality result and we develop an algorithm
with provable numerical convergence. Furthermore, we demonstrate the
performance of the method on simulated and a real high-dimensional dataset.

Large contingency tables summarizing categorical variables arise in many
areas. For example in biology when a large number of biomarkers are
cross-tabulated according to their discrete expression level. Interactions of
the variables are generally studied with log-linear models and the structure of
a log-linear model can be visually represented by a graph from which the
conditional independence structure can then be read off.

We consider variable selection in high-dimensional linear models where the
number of covariates greatly exceeds the sample size. We introduce the new
concept of partial faithfulness and use it to infer associations between the
covariates and the response.

Oracle inequalities and variable selection properties for the Lasso in linear
models have been established under a variety of different assumptions on the
design matrix. We show in this paper how the different conditions and concepts
relate to each other. The restricted eigenvalue condition (Bickel et al., 2009)
or the slightly weaker compatibility condition (van de Geer, 2007) are
sufficient for oracle results. We argue that both these conditions allow for a
fairly general class of design matrices.