This paper discusses asymptotically distribution free tests for the classical
goodness-of-fit hypothesis of an error distribution in nonparametric regression
models. These tests are based on the same martingale transform of the residual
empirical process as used in the one sample location model. This transformation
eliminates extra randomization due to covariates but not due the errors, which
is intrinsically present in the estimators of the regression function. Thus,
tests based on the transformed process have, generally, better power.

We suggest a robust nearest-neighbor approach to classifying high-dimensional
data. The method enhances sensitivity by employing a threshold and truncates to
a sequence of zeros and ones in order to reduce the deleterious impact of
heavy-tailed data. Empirical rules are suggested for choosing the threshold.
They require the bare minimum of data; only one data vector is needed from each
population. Theoretical and numerical aspects of performance are explored,
paying particular attention to the impacts of correlation and heterogeneity
among data components.

The p_1 model is a directed random graph model used to describe dyadic
interactions in a social network in terms of effects due to differential
attraction (popularity) and expansiveness, as well as an additional effect due
to reciprocation. In this article we carry out an algebraic statistics analysis
of this model. We show that the p_1 model is a toric model specified by a
multi-homogeneous ideal. We conduct an extensive study of the Markov bases for
p_1 models that incorporate explicitly the constraint arising from
multi-homogeneity.

Let $p_n(y)=\sum_k\hat{\alpha}_k\phi(y-k)+\sum_{l=0}^{j_n-1}\sum_k\hat
{\beta}_{lk}2^{l/2}\psi(2^ly-k)$ be the linear wavelet density estimator, where
$\phi$, $\psi$ are a father and a mother wavelet (with compact support),
$\hat{\alpha}_k$, $\hat{\beta}_{lk}$ are the empirical wavelet coefficients
based on an i.i.d.

We discuss connecting tables with zero-one entries by a subset of a Markov
basis. In this paper, as a Markov basis we consider the Graver basis, which
corresponds to the unique minimal Markov basis for the Lawrence lifting of the
original configuration. Since the Graver basis tends to be large, it is of
interest to clarify conditions such that a subset of the Graver basis, in
particular a minimal Markov basis itself, connects tables with zero-one
entries. We give some theoretical results on the connectivity of tables with
zero-one entries.

This is a technical appendix to "Adaptive estimation of stationary Gaussian
fields". We present several proofs that have been skipped in the main paper.

We reconsider the existing kernel estimators for a copula function, as
proposed in Gijbels and Mielniczuk [Comm. Statist. Theory Methods 19 (1990)
445--464], Fermanian, Radulovi\v{c} and Wegkamp [Bernoulli 10 (2004) 847--860]
and Chen and Huang [Canad. J. Statist. 35 (2007) 265--282]. All of these
estimators have as a drawback that they can suffer from a corner bias problem.
A way to deal with this is to impose rather stringent conditions on the copula,
outruling as such many classical families of copulas.

In this manuscript we introduce a generalisation of the log-Normal
distribution that is inspired by a modification of the Kaypten multiplicative
process using the $q$-product of Borges [Physica A \textbf{340}, 95 (2004)].
Depending on the value of q the distribution increases the tail for small (when
$q<1$) or large (when $q>1$) values of the variable upon analysis. The usual
log-Normal distribution is retrieved when $q=1$. The main statistical features
of this distribution are presented as well as a related random number
generators and tables of quantiles of the Kolmogorov-Smirnov.

In this paper, we study the asymptotic posterior distribution of linear
functionals of the density. In particular, we give general conditions to obtain
a semiparametric version of the Bernstein-Von Mises theorem. We then apply this
general result to nonparametric priors based on infinite dimensional
exponential families. As a byproduct, we also derive adaptive nonparametric
rates of concentration of the posterior distributions under these families of
priors on the class of Sobolev and Besov spaces.

Principal component analysis (PCA) is a classical method for dimensionality
reduction based on extracting the dominant eigenvectors of the sample
covariance matrix. However, PCA is well known to behave poorly in the ``large
$p$, small $n$'' setting, in which the problem dimension $p$ is comparable to
or larger than the sample size $n$. This paper studies PCA in this
high-dimensional regime, but under the additional assumption that the maximal
eigenvector is sparse, say, with at most $k$ nonzero components.

Consider a continuous random pair $(X,Y)$ whose dependence is characterized
by an extreme-value copula with Pickands dependence function $A$. When the
marginal distributions of $X$ and $Y$ are known, several consistent estimators
of $A$ are available. Most of them are variants of the estimators due to
Pickands [Bull. Inst. Internat. Statist. 49 (1981) 859--878] and
Cap\'{e}ra\`{a}, Foug\`{e}res and Genest [Biometrika 84 (1997) 567--577]. In
this paper, rank-based versions of these estimators are proposed for the more
common case where the margins of $X$ and $Y$ are unknown.

In arXiv:0907.0079 by Cator and Lopuhaa, an asymptotic expansion for the MCD
estimators is established in a very general framework. This expansion requires
the existence and non-singularity of the derivative in a first-order Taylor
expansion. In this paper, we prove the existence of this derivative for
multivariate distributions that have a density and provide an explicit
expression. Moreover, under suitable symmetry conditions on the density, we
show that this derivative is non-singular.

In 1985, for detecting changes in distributions Pollak introduced a specific
minimax performance metric and a randomized version of the Shiryaev-Roberts
procedure where the zero initial condition is replaced by a random variable
sampled from the quasi-stationary distribution. Pollak proved that this
procedure is third-order asymptotically optimal as the mean time to false alarm
becomes large. The question whether Pollak's procedure is strictly minimax for
any false alarm rate has been open for more than two decades, and there were
several attempts to prove this strict optimality.

We give a finite-sample analysis of predictive inference procedures after
model selection in regression with random design. The analysis is focused on a
statistically challenging scenario where the number of potentially important
explanatory variables can be infinite, where no regularity conditions are
imposed on unknown parameters, where the number of explanatory variables in a
"good" model can be of the same order as sample size and where the number of
candidate models can be of larger order than sample size.

Stochastic approximation Monte Carlo (SAMC) has recently been proposed by
Liang, Liu and Carroll [J. Amer. Statist. Assoc. 102 (2007) 305--320] as a
general simulation and optimization algorithm. In this paper, we propose to
improve its convergence using smoothing methods and discuss the application of
the new algorithm to Bayesian model selection problems. The new algorithm is
tested through a change-point identification example. The numerical results
indicate that the new algorithm can outperform SAMC and reversible jump MCMC
significantly for the model selection problems.

Consider the problem of estimating the $\gamma$-level set
$G^*_{\gamma}=\{x:f(x)\geq\gamma\}$ of an unknown $d$-dimensional density
function $f$ based on $n$ independent observations $X_1,...,X_n$ from the
density. This problem has been addressed under global error criteria related to
the symmetric set difference. However, in certain applications a spatially
uniform mode of convergence is desirable to ensure that the estimated set is
close to the target set everywhere. The Hausdorff error criterion provides this
degree of uniformity and, hence, is more appropriate in such situations.

We consider estimation of quantile curves for a general class of
nonstationary processes. Consistency and central limit results are obtained for
local linear quantile estimates under a mild short-range dependence condition.
Our results are applied to environmental data sets. In particular, our results
can be used to address the problem of whether climate variability has changed,
an important problem raised by IPCC (Intergovernmental Panel on Climate Change)
in 2001.

Bayesian and frequentist methods differ in many aspects, but share some basic
optimality properties. In practice, there are situations in which one of the
methods is more preferred by some criteria. We consider the case of inference
about a set of multiple parameters, which can be divided into two disjoint
subsets. On one set, a frequentist method may be favored and on the other, the
Bayesian.

In this paper we are interested in empirical likelihood (EL) as a method of
estimation, and we address the following two problems: (1) selecting among
various empirical discrepancies in an EL framework and (2) demonstrating that
EL has a well-defined probabilistic interpretation that would justify its use
in a Bayesian context. Using the large deviations approach, a Bayesian law of
large numbers is developed that implies that EL and the Bayesian maximum a
posteriori probability (MAP) estimators are consistent under misspecification
and that EL can be viewed as an asymptotic form of MAP.

We consider tests of hypotheses when the parameters are not identifiable
under the null in semiparametric models, where regularity conditions for
profile likelihood theory fail. Exponential average tests based on integrated
profile likelihood are constructed and shown to be asymptotically optimal under
a weighted average power criterion with respect to a prior on the
nonidentifiable aspect of the model. These results extend existing results for
parametric models, which involve more restrictive assumptions on the form of
the alternative than do our results.

The research of developing a general methodology for the construction of good
nonregular designs has been very active in the last decade. Recent research by
Xu and Wong [Statist. Sinica 17 (2007) 1191--1213] suggested a new class of
nonregular designs constructed from quaternary codes. This paper explores the
properties and uses of quaternary codes toward the construction of
quarter-fraction nonregular designs. Some theoretical results are obtained
regarding the aliasing structure of such designs.

Normal mixture distributions are arguably the most important mixture models,
and also the most technically challenging. The likelihood function of the
normal mixture model is unbounded based on a set of random samples, unless an
artificial bound is placed on its component variance parameter. Moreover, the
model is not strongly identifiable so it is hard to differentiate between over
dispersion caused by the presence of a mixture and that caused by a large
variance, and it has infinite Fisher information with respect to mixing
proportions.

We study a class of hypothesis testing problems in which, upon observing the
realization of an n-dimensional Gaussian vector, one has to decide whether the
vector was drawn from a standard normal distribution or, alternatively, whether
there is a subset of the components belonging to a certain given class of sets
whose elements have been "contaminated," that is, have a mean different from
zero. We establish some general conditions under which testing is possible and
others under which testing is hopeless with a small risk.

This paper proves fixed domain asymptotic results for estimating a smooth
invertible transformation $f:\Bbb{R}^2\to\Bbb{R}^2$ when observing the deformed
random field $Z\circ f$ on a dense grid in a bounded, simply connected domain
$\Omega$, where $Z$ is assumed to be an isotropic Gaussian random field on
$\Bbb{R}^2$. The estimate $\hat{f}$ is constructed on a simply connected domain
$U$, such that $\overline{U}\subset\Omega$ and is defined using kernel smoothed
quadratic variations, Bergman projections and results from quasiconformal
theory.

Given a sample from a discretely observed L\'evy process $X=(X_t)_{t\geq 0}$
of the finite jump activity, we study the problem of nonparametric estimation
of the L\'evy density $\rho$ corresponding to the process $X.$ Our estimator of
$\rho$ is based on a suitable inversion of the L\'evy-Khintchine formula and a
plug-in device. The main result of the paper deals with an upper bound on the
mean square error of the estimator of $\rho$ at a fixed point $x.$ We also show
that the estimator attains the minimax convergence rate over a suitable class
of L\'evy densities.

We define a generalized index of jump activity, propose estimators of that
index for a discretely sampled process and derive the estimators' properties.
These estimators are applicable despite the presence of Brownian volatility in
the process, which makes it more challenging to infer the characteristics of
the small, infinite activity jumps. When the method is applied to high
frequency stock returns, we find evidence of infinitely active jumps in the
data and estimate their index of activity.

The problem we concentrate on is as follows: given (1) a convex compact set
$X$ in ${\mathbb{R}}^n$, an affine mapping $x\mapsto A(x)$, a parametric family
$\{p_{\mu}(\cdot)\}$ of probability densities and (2) $N$ i.i.d. observations
of the random variable $\omega$, distributed with the density $p_{A(x)}(\cdot)$
for some (unknown) $x\in X$, estimate the value $g^Tx$ of a given linear form
at $x$.

Regression models to relate a scalar $Y$ to a functional predictor $X(t)$ are
becoming increasingly common. Work in this area has concentrated on estimating
a coefficient function, $\beta(t)$, with $Y$ related to $X(t)$ through
$\int\beta(t)X(t) dt$. Regions where $\beta(t)\ne0$ correspond to places where
there is a relationship between $X(t)$ and $Y$. Alternatively, points where
$\beta(t)=0$ indicate no relationship.

We derive sharp performance bounds for least squares regression with $L_1$
regularization from parameter estimation accuracy and feature selection quality
perspectives. The main result proved for $L_1$ regularization extends a similar
result in [Ann. Statist. 35 (2007) 2313--2351] for the Dantzig selector. It
gives an affirmative answer to an open question in [Ann. Statist. 35 (2007)
2358--2364]. Moreover, the result leads to an extended view of feature
selection that allows less restrictive conditions than some recent work.

Let M be a smooth compact oriented manifold without boundary, imbedded in a
euclidean space E and let f be a smooth map of M into a Riemannian manifold N.
An unknown state x in M is observed via X=x+su where s>0 is a small parameter
and u is a white Gaussian noise. For a given smooth prior on M and smooth
estimators g of the map f we have derived a second-order asymptotic expansion
for the related Bayesian risk (see arXiv:0705.2540). In this paper, we apply
this technique to a variety of examples.