Jon A Wellner

A New Approach to Tests and Confidence Bands for Distribution Functions
Lutz Duembgen, Jon A. Wellner
We introduce new goodness-of-fit tests and corresponding confidence bands for distribution functions. They are inspired by multi-scale methods of testing and…

Hardy's Inequality and Its Descendants
Chris A. J. Klaassen, Jon A. Wellner
We formulate and prove a generalization of Hardy's inequality (Hardy,1925) in terms of random variables and show that it contains the usual (or familiar)…

Bi-$s^*$-Concave Distributions
Nilanjana Laha, Zhen Miao, Jon A. Wellner
We introduce new shape-constrained classes of distribution functions on R, the bi-$s^*$-concave classes. In parallel to results of Dümbgen, Kolesnyk, and Wilke…

Estimation of mean residual life
W. J. Hall, Jon A. Wellner
Yang (1978) considered an empirical estimate of the mean residual life function on a fixed finite interval. She proved it to be strongly uniformly consistent…

Bi-$s^*$-concave distributions
Nilanjana Laha, Jon A. Wellner
We introduce a new shape-constrained class of distribution functions on R, the bi-$s^*$-concave class. In parallel to results of Dümbgen, Kolesnyk, and Wilke …

The Bennett-Orlicz norm
Jon A. Wellner
Lederer and van de Geer (2013) introduced a new Orlicz norm, the Bernstein-Orlicz norm, which is connected to Bernstein type inequalities. Here we introduce…

Entropy of convex functions on $R^d$
Fuchang Gao, Jon A. Wellner
Let $\Omega$ be a bounded closed convex set in ${\mathbb R}^d$ with non-empty interior, and let ${\cal C}_r(\Omega)$ be the class of convex functions on $…

Finite sampling inequalities: an application to two-sample Kolmogorov-Smirnov statistics
Evan Greene, Jon A. Wellner
We review a finite-sampling exponential bound due to Serfling and discuss related exponential bounds for the hypergeometric distribution. We then discuss how…

Exponential bounds for the hypergeometric distribution
Evan Greene, Jon A. Wellner
We establish exponential bounds for the hypergeometric distribution which include a finite sampling correction factor, but are otherwise analogous to bounds…

Multivariate convex regression: global risk bounds and adaptation
Qiyang Han, Jon A. Wellner
We study the problem of estimating a multivariate convex function defined on a convex body in a regression setting with random design. We are interested in…

Approximation and Estimation of s-Concave Densities via Rényi Divergences
Qiyang Han, Jon A. Wellner
In this paper, we study the approximation and estimation of $s$-concave densities via Rényi divergence. We first show that the approximation of a probability…

Global Rates of Convergence of the MLEs of Log-concave and s-concave Densities
Charles R. Doss, Jon A. Wellner
We establish global rates of convergence for the Maximum Likelihood Estimators (MLEs) of log-concave and $s$-concave densities on $\mathbb{R}$. The main…

A law of the iterated logarithm for Grenander's estimator
Lutz Duembgen, Jon A. Wellner, Malcolm Wolff
In this note we prove the following law of the iterated logarithm for the Grenander estimator of a monotone decreasing density: If $f(t_0) > 0$, $f'(t_0) 0$,…

An excursion approach to maxima of the Brownian Bridge
Mihael Perman, Jon A. Wellner
Functionals of Brownian bridge arise as limiting distributions in nonparametric statistics. In this paper we will give a derivation of distributions of extrema…

Log-concavity and strong log-concavity: a review
Adrien Saumard, Jon A. Wellner
We review and formulate results concerning log-concavity and strong-log-concavity in both discrete and continuous settings. We show how preservation of log…

Chernoff's density is log-concave
Fadoua Balabdaoui, Jon A. Wellner
We show that the density of $Z=\mathop {\operatorname {argmax}}\{W(t)-t^2\}$, sometimes known as Chernoff's density, is log-concave. We conjecture that…

Weighted likelihood estimation under two-phase sampling
Takumi Saegusa, Jon A. Wellner
We develop asymptotic theory for weighted likelihood estimators (WLE) under two-phase stratified sampling without replacement. We also consider several…

On the Hermite spline conjecture and its connection to k-monotone densities
Fadoua Balabdaoui, Simon Foucart, Jon A. Wellner
The k-monotone classes of densities defined on (0, ∞) have been known in the mathematical literature but were for the first time considered from a…

Global Rates of Convergence of the MLE for Multivariate Interval Censoring
Jon A. Wellner, Fuchang Gao
We establish global rates of convergence of the Maximum Likelihood Estimator (MLE) of a multivariate distribution function in the case of (one type of) …

Nonparametric estimation of multivariate convex-transformed densities
Arseni Seregin, Jon A. Wellner
We study estimation of multivariate densities $p$ of the form $p(x)=h(g(x))$ for $x\in \mathbb {R}^d$ and for a fixed monotone function $h$ and an unknown…

Squaring the Circle and Cubing the Sphere: Circular and Spherical Copulas
Michael D. Perlman, Jon A. Wellner
Do there exist circular and spherical copulas in $R^d$? That is, do there exist circularly symmetric distributions on the unit disk in $R^2$ and spherically…

A local maximal inequality under uniform entropy
Aad van der Vaart, Jon A. Wellner
We derive an upper bound for the mean of the supremum of the empirical process indexed by a class of functions that are known to have variance bounded by a…

Nonparametric estimation of multivariate scale mixtures of uniform densities
Marios G. Pavlides, Jon A. Wellner
Suppose that $\m{U} = (U_1, \ldots , U_d) $ has a Uniform$([0,1]^d)$ distribution, that $\m{Y} = (Y_1 , \ldots , Y_d) $ has the distribution $G$ on $\RR_+^d$,…

Nonparametric estimation of a convex bathtub-shaped hazard function
Hanna K. Jankowski, Jon A. Wellner
In this paper, we study the nonparametric maximum likelihood estimator (MLE) of a convex hazard function. We show that the MLE is consistent and converges at a…

How many Laplace transforms of probability measures are there?
Fuchang Gao, Wenbo V. Li, Jon A. Wellner
A bracketing metric entropy bound for the class of Laplace transforms of probability measures on [0,∞) is obtained through its connection with the small…

Estimation of a discrete monotone distribution
Hanna K. Jankowski, Jon A. Wellner
We study and compare three estimators of a discrete monotone distribution: (a) the (raw) empirical estimator; (b) the "method of rearrangements" estimator; and…

On the Grenander estimator at zero
Fadoua Balabdaoui, Hanna K. Jankowski, Marios Pavlides, Arseni Seregin, Jon A. Wellner
We establish limit theory for the Grenander estimator of a monotone density near zero. In particular we consider the situation when the true density $f_0$ is…

Nemirovski's Inequalities Revisited
Lutz Duembgen, Sara van de Geer, Mark Veraar, Jon A. Wellner
An important tool for statistical research are moment inequalities for sums of independent random vectors. Nemirovski and coworkers (1983, 2000) derived one…

Inconsistency of the MLE for the joint distribution of interval censored survival times and continuous marks
Marloes H. Maathuis, Jon A. Wellner
This paper considers the nonparametric maximum likelihood estimator (MLE) for the joint distribution function of an interval censored survival time and a…

How many distribution functions are there? Bracketing entropy bounds for high dimensional distribution functions
Shuguang Song, Jon A. Wellner
This paper has been withdrawn by the authors due to a crucial error in a bound on page 19 and some other errors earlier in the paper.

Estimation of a $k$-monotone density: limit distribution theory and the spline connection
Fadoua Balabdaoui, Jon A. Wellner
We study the asymptotic behavior of the Maximum Likelihood and Least Squares Estimators of a $k$-monotone density $g_0$ at a fixed point $x_0$ when $k>2$. We…

Two likelihood-based semiparametric estimation methods for panel count data with covariates
Jon A. Wellner, Ying Zhang
We consider estimation in a particular semiparametric regression model for the mean of a counting process with ``panel count'' data. The basic model assumption…

Goodness-of-fit tests via phi-divergences
Leah Jager, Jon A. Wellner
A unified family of goodness-of-fit tests based on $\phi$-divergences is introduced and studied. The new family of test statistics $S_n(s)$ includes both the…

A Kiefer--Wolfowitz theorem for convex densities
Fadoua Balabdaoui, Jon A. Wellner
Kiefer and Wolfowitz [Z. Wahrsch. Verw. Gebiete 34 (1976) 73--85] showed that if $F$ is a strictly curved concave distribution function (corresponding to a…

Entropy Estimate For High Dimensional Monotonic Functions
Fuchang Gao, Jon A. Wellner
We establish upper and lower bounds for the metric entropy and bracketing entropy of the class of $d$-dimensional bounded monotonic functions under $L^p$ norms…

Weighted Likelihood for Semiparametric Models and Two-phase Stratified Samples, with Application to Cox Regression
Norman E. Breslow, Jon A. Wellner
Weighted likelihood, in which one solves Horvitz-Thompson or inverse probability weighted (IPW) versions of the likelihood equations, offers a simple and…

Conjecture of error boundedness in a new Hermite interpolation problem via splines of odd-degree
Fadoua Balabdaoui, Jon A. Wellner
We present a Hermite interpolation problem via splines of odd-degree which, to the best knowledge of the authors, has not been considered in the literature on…

Estimation of a k-monotone density, part 1: characterizations consistency, and minimax lower bounds
Fadoua Balabdaoui, Jon A. Wellner
Shape constrained densities are encountered in many nonparametric estimation problems. The classes of monotone or convex (and monotone) densities can be viewed…

The suppport reduction algorithm for computing nonparametric function estimates in mixture models
Piet Groeneboom, Geurt Jongbloed, Jon A. Wellner
Vertex direction algorithms have been around for a few decades in the experimental design and mixture models literature. We briefly review this type of…