Most papers on high-dimensional statistics are based on the assumption that
none of the regressors are correlated with the regression error, namely, they
are exogeneous. Yet, endogeneity arises easily in high-dimensional regression
due to a large pool of regressors and this causes the inconsistency of the
penalized least-squares methods and possible false scientific discoveries. A
necessary condition for model selection of a very general class of penalized
regression methods is given, which allows us to prove formally the
inconsistency claim.

The multiple testing procedure plays an important role in detecting the
presence of spatial signals for large-scale imaging data. Typically, the
spatial signals are sparse but clustered.

Multiple hypothesis testing is a fundamental problem in high dimensional
inference, with wide applications in many scientific fields. In genome-wide
association studies, tens of thousands of tests are performed simultaneously to
find if any genes are associated with some traits and those tests are
correlated. When test statistics are correlated, false discovery control
becomes very challenging under arbitrary dependence.

We propose several statistics to test the Markov hypothesis for
$\beta$-mixing stationary processes sampled at discrete time intervals. Our
tests are based on the Chapman--Kolmogorov equation. We establish the
asymptotic null distributions of the proposed test statistics, showing that
Wilks's phenomenon holds. We compute the power of the test and provide
simulations to investigate the finite sample performance of the test statistics
when the null model is a diffusion process, with alternatives consisting of
models with a stochastic mean reversion level, stochastic volatility and jumps.

Multiple hypothesis testing is a fundamental problem in high dimensional
inference, with wide applications in many scientific fields. In genome-wide
association studies, tens of thousands of tests are performed simultaneously to
find if any genes are associated with some traits and those tests are
correlated. When test statistics are correlated, false discovery control
becomes very challenging under arbitrary dependence.

Motivated by normalizing DNA microarray data and by predicting the interest
rates, we explore nonparametric estimation of additive models with highly
correlated covariates. We introduce two novel approaches for estimating the
additive components, integration estimation and pooled backfitting estimation.
The former is designed for highly correlated covariates, and the latter is
useful for nonhighly correlated covariates. Asymptotic normalities of the
proposed estimators are established.

Variable selection in high dimensional space has challenged many contemporary
statistical problems from many frontiers of scientific disciplines. Recent
technology advance has made it possible to collect a huge amount of covariate
information such as microarray, proteomic and SNP data via bioimaging
technology while observing survival information on patients in clinical
studies. Thus, the same challenge applies to the survival analysis in order to
understand the association between genomics information and clinical
information about the survival time.

In high-dimensional model selection problems, penalized simple least-square
approaches have been extensively used. This paper addresses the question of
both robustness and efficiency of penalized model selection methods, and
proposes a data-driven weighted linear combination of convex loss functions,
together with weighted $L_1$-penalty. It is completely data-adaptive and does
not require prior knowledge of the error distribution. The weighted
$L_1$-penalty is used both to ensure the convexity of the penalty term and to
ameliorate the bias caused by the $L_1$-penalty.

This paper studies the sparsistency and rates of convergence for estimating
sparse covariance and precision matrices based on penalized likelihood with
nonconvex penalty functions. Here, sparsistency refers to the property that all
parameters that are zero are actually estimated as zero with probability
tending to one. Depending on the case of applications, sparsity priori may
occur on the covariance matrix, its inverse or its Cholesky decomposition. We
study these three sparsity exploration problems under a unified framework with
a general penalty function.

Ultrahigh dimensional variable selection plays an increasingly important role
in contemporary scientific discoveries and statistical research. Among others,
Fan and Lv (2008) propose an independent screening framework by ranking the
marginal correlations. They showed that the correlation ranking procedure
possesses a sure independence screening property within the context of the
linear model with Gaussian covariates and responses.

High dimensional statistical problems arise from diverse fields of scientific
research and technological development. Variable selection plays a pivotal role
in contemporary statistical learning and scientific discoveries. The
traditional idea of best subset selection methods, which can be regarded as a
specific form of penalized likelihood, is computationally too expensive for
many modern statistical applications. Other forms of penalized likelihood
methods have been successfully developed over the last decade to cope with high
dimensionality.

Penalized likelihood methods are fundamental to ultra-high dimensional
variable selection. How high dimensionality such methods can handle remains
largely unknown. In this paper, we show that in the context of generalized
linear models, such methods possess model selection consistency with oracle
properties even for dimensionality of Non-Polynomial (NP) order of sample size,
for a class of penalized likelihood approaches using folded-concave penalty
functions, which were introduced to ameliorate the bias problems of convex
penalty functions.