# Jon A Wellner

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

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Hardy's Inequality and Its Descendants
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Chris A. J. Klaassen, Jon A. Wellner
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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)…

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Bi-$s^*$-Concave Distributions
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Nilanjana Laha, Zhen Miao, Jon A. Wellner
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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…

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Estimation of mean residual life
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W. J. Hall, Jon A. Wellner
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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…

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Bi-$s^*$-concave distributions
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Nilanjana Laha, Jon A. Wellner
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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 …

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The Bennett-Orlicz norm
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Jon A. Wellner
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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…

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Entropy of convex functions on $R^d$
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Fuchang Gao, Jon A. Wellner
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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 $…

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

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

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Multivariate convex regression: global risk bounds and adaptation
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Qiyang Han, Jon A. Wellner
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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…

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Approximation and Estimation of s-Concave Densities via Rényi Divergences
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Qiyang Han, Jon A. Wellner
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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…

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

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

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

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

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Chernoff's density is log-concave
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Fadoua Balabdaoui, Jon A. Wellner
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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…

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

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On the Hermite spline conjecture and its connection to k-monotone densities
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Fadoua Balabdaoui, Simon Foucart, Jon A. Wellner
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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…

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

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Nonparametric estimation of multivariate convex-transformed densities
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Arseni Seregin, Jon A. Wellner
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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…

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Squaring the Circle and Cubing the Sphere: Circular and Spherical Copulas
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Michael D. Perlman, Jon A. Wellner
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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…

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A local maximal inequality under uniform entropy
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Aad van der Vaart, Jon A. Wellner
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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…

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Nonparametric estimation of multivariate scale mixtures of uniform densities
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Marios G. Pavlides, Jon A. Wellner
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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 $\RR_+^d$,…

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Nonparametric estimation of a convex bathtub-shaped hazard function
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Hanna K. Jankowski, Jon A. Wellner
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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…

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

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

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On the Grenander estimator at zero
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Fadoua Balabdaoui, Hanna K. Jankowski, Marios Pavlides, Arseni Seregin, Jon A. Wellner
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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…

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

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

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How many distribution functions are there? Bracketing entropy bounds for high dimensional distribution functions
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Shuguang Song, Jon A. Wellner
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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.

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Estimation of a $k$-monotone density: limit distribution theory and the spline connection
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Fadoua Balabdaoui, Jon A. Wellner
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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…

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

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Goodness-of-fit tests via phi-divergences
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Leah Jager, Jon A. Wellner
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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…

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A Kiefer--Wolfowitz theorem for convex densities
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Fadoua Balabdaoui, Jon A. Wellner
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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…

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Entropy Estimate For High Dimensional Monotonic Functions
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Fuchang Gao, Jon A. Wellner
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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…

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

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Conjecture of error boundedness in a new Hermite interpolation problem via splines of odd-degree
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Fadoua Balabdaoui, Jon A. Wellner
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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…

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

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