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Jon A. Wellner

University of Washington, Statistics, Faculty Member

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Papers by Jon A. Wellner

A law of the iterated logarithm for Grenander’s estimator

Stochastic Processes and their Applications, 2016

In this note we prove the following law of the iterated logarithm for the Grenander estimator of ... more 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, and f is continuous in a neighborhood of t 0 , then lim sup n→∞ n 2 log log n 1/3 (fn(t 0) − f (t 0)) = f (t 0)f (t 0)/2 1/3 2M almost surely where M ≡ sup g∈G Tg = (3/4) 1/3 and Tg ≡ argmax u {g(u) − u 2 }; here G is the two-sided Strassen limit set on R. The proof relies on laws of the iterated logarithm for local empirical processes, Groeneboom's switching relation, and properties of Strassen's limit set analogous to distributional properties of Brownian motion.

format_quoteProves a Law of the Iterated Logarithm (LIL) for Grenander estimator, providing key results for monotone decreasing densities.format_quote

Marshall’s lemma for convex density estimation

Institute of Mathematical Statistics Lecture Notes - Monograph Series, 2007

Marshall's [Nonparametric Techniques in Statistical Inference (1970) 174-176] lemma is an analyti... more Marshall's [Nonparametric Techniques in Statistical Inference (1970) 174-176] lemma is an analytical result which implies √ n-consistency of the distribution function corresponding to the Grenander [Skand. Aktuarietidskr. 39 (1956) 125-153] estimator of a non-decreasing probability density. The present paper derives analogous results for the setting of convex densities on [0, ∞).

Chernoff’s density is log-concave

Bernoulli, 2014

We show that the density of Z = argmax{W (t) − t 2 }, sometimes known as Chernoff's density, is l... more

format_quoteNumerical computations for Chernoff's density and its moments were provided, enhancing its practical understanding.format_quote

A Kiefer-Wolfowitz theorem for convex densities

Institute of Mathematical Statistics Lecture Notes - Monograph Series, 2007

Kiefer and Wolfowitz [Z. Wahrsch. Verw. Gebiete 34 (1976) 73-85] showed that if F is a strictly c... more 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 strictly monotone density f), then the Maximum Likelihood Estimator Fn, which is, in fact, the least concave majorant of the empirical distribution function Fn, differs from the empirical distribution function in the uniform norm by no more than a constant times (n −1 log n) 2/3 almost surely. We review their result and give an updated version of their proof. We prove a comparable theorem for the class of distribution functions F with convex decreasing densities f , but with the maximum likelihood estimator Fn of F replaced by the least squares estimator Fn: if X 1 ,. .. , Xn are sampled from a distribution function F with strictly convex density f , then the least squares estimator Fn of F and the empirical distribution function Fn differ in the uniform norm by no more than a constant times (n −1 log n) 3/5 almost surely. The proofs rely on bounds on the interpolation error for complete spline interpolation due to Hall [

On the Grenander estimator at zero

Statistica Sinica, 2011

We establish limit theory for the Grenander estimator of a monotone density near zero. In particu... more We establish limit theory for the Grenander estimator of a monotone density near zero. In particular we consider the situation when the true density f0 is unbounded at zero, with different rates of growth to infinity. In the course of our study we develop new switching relations by use of tools from convex analysis. The theory is applied to a problem involving mixtures.

Nemirovski's Inequalities Revisited

An important tool for statistical research are moment inequalities for sums of independent random... more An important tool for statistical research are moment inequalities for sums of independent random vectors. Nemirovski and coworkers (1983, 2000) derived one particular type of such inequalities: For certain Banach spaces (,·) there exists a constant K = K(,·) such that for arbitrary independent and centered random vectors X_1, X_2, ..., X_n ∈, their sum S_n satisfies the inequality E S_n ^2 < K ∑_i=1^n E X_i^2. We present and compare three different approaches to obtain such inequalities: Nemirovski's results are based on deterministic inequalities for norms. Another possible vehicle are type and cotype inequalities, a tool from probability theory on Banach spaces. Finally, we use a truncation argument plus Bernstein's inequality to obtain another version of the moment inequality above. Interestingly, all three approaches have their own merits.

format_quoteCurrent research shows significant dimension effects in statistical inequalities, impacting sample sizes vs. measured items.format_quote

Log-concavity and strong log-concavity: a review

We review and formulate results concerning log-concavity and strong-log-concavity in both discret... more We review and formulate results concerning log-concavity and strong-log-concavity in both discrete and continuous settings. We show how preservation of log-concavity and strongly log-concavity on R under convolution follows from a fundamental monotonicity result of Efron (1969). We provide a new proof of Efron's theorem using the recent asymmetric Brascamp-Lieb inequality due to Otto and Menz (2013). Along the way we review connections between log-concavity and other areas of mathematics and statistics, including concentration of measure, log-Sobolev inequalities, convex geometry, MCMC algorithms, Laplace approximations, and machine learning.

$Fic 1. Sherman’s example, h = px q It is also wor dimension and t th noting the following difference between the situation in one he result for preservation of convolution in higher dimensions: as we note following Theorems 29 and 33, Ibragimov [1956a] and Keilson and Ger- ber [1971] showed that in the one-dimensional continuous and discrete settings respectively tha t if pxq is unimodal for every unimodal gq, then p is log-concave. The analogue of this for d > 1 is more complicated in part because of the great variety of possi and Joag-Dev [I Sherman [1955] ble definitions of “unimodal” in this case; see Dharmadhikari 988] chapters 2 and 3 for a thorough discussion. In particular provided the following counterexample when the notion of uni- modality is taken to be centrally symmetric convex unimodality; that is, the sets S.(p) = {x € R?: p(x) > c} are symmetric and convex for each c > 0. Let p be the uniform density on [—1, 1]? (so that p(x) = (1/4)1,~1,1)2(«)); then p is log-concave ({-1, 1] x (1, 5]) (and hence also . Let g be the density given by 1/12 on [—1,1]? and 1/24 on U ([-1,1] x [—5,-1)). Thus q is centrally symmetric convex quasi-concave, q € P_oo as in Definition 2.5. But h = pxq is not centrally symmetric convex (and also is not quasi-concave), since the sets S.(h) are not convex: see Figure 1.$

Efficient and Adaptive Estimation in Non-and Semiparametric Models

Marshall’s lemma for convex density estimation

by Lutz Duembgen and Jon A. Wellner

Institute of Mathematical Statistics Lecture Notes - Monograph Series, 2007

lemma is an analytical result which implies √ n-consistency of the distribution function correspo... more lemma is an analytical result which implies √ n-consistency of the distribution function corresponding to the Grenander [Skand. Aktuarietidskr. 39 (1956) 125-153] estimator of a non-decreasing probability density. The present paper derives analogous results for the setting of convex densities on [0, ∞).

format_quoteIntroduces √n-consistency results for convex density estimators, highlighting their relevance compared to Grenander's non-decreasing case.format_quote

Efficient and Adaptive Estimation for Semiparametric Models

by Chris Klaassen and Jon A. Wellner

Biometrics, 1994

Peter Bickel, Chris AJ Klaassen, Ya&amp;amp;amp;#x27;acov Ritov, and Jon A. Wellner ... 2. As... more Peter Bickel, Chris AJ Klaassen, Ya&amp;amp;amp;#x27;acov Ritov, and Jon A. Wellner ... 2. Asymptotic Inference for (Finite-Dimensional) Parametric Models ... 2.1 Regular parametric models in the iid case :::::::::::::::::::::::::::::::::::11 ... 2.3 The information bound and the H&amp;amp;amp;#x27;ajek - Le Cam convolution and asymptotic minimax theorems ::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 23 ... 2.4 Nuisance parameters, adaptation, and some geometry ::::::::::::::::::::::: 27 ... 2.5 Construction of p n consistent and e cient estimates :::::::::::::::::::::: 40 ... 3. Information Bounds for Euclidean Parameters ...

Mixtures and Monotonicity: a Review of Estimation under Monotonicity Constraints

by Fadoua Balabdaoui and Jon A. Wellner

Monotone and multiply monotone densities have well-known mix- ture representations. The underlyin... more Monotone and multiply monotone densities have well-known mix- ture representations. The underlying mixture representations give rise to a wide variety of fascinating inverse problems. We review the forward problems (estimation of the mixed density) and the inverse problems (estimation of the mixing distribution in two different guises), including Hampel's bird resting time problem and generalizations thereof. Section 5 gives a

format_quoteThe type B representation shows how k-monotone densities can be expressed, offering insights into biased distribution relationships.format_quote

On the Grenander estimator at zero

by Fadoua Balabdaoui and Jon A. Wellner

Statistica Sinica, 2011

We establish limit theory for the Grenander estimator of a monotone density near zero. In particu... more We establish limit theory for the Grenander estimator of a monotone density near zero. In particular we consider the situation when the true density f0 is unbounded at zero, with different rates of growth to infinity. In the course of our study we develop new switching relations by use of tools from convex analysis. The theory is applied to a problem involving mixtures.

Iterative Spline Algorithms for Estimating a k-monotone Density

by Fadoua Balabdaoui and Jon A. Wellner

The iterative (2k− 1)-st spline algorithm is an extension of the iterative cubic spline algorithm... more The iterative (2k− 1)-st spline algorithm is an extension of the iterative cubic spline algorithm developed and used by Groeneboom, Jongbloed and Wellner to compute an approximation of the “invelope” of the integrated two-sided Brownian motion+ t4 that is involved in the limiting distribution of the MLE or the LSE of a non-increasing and convex density on (0,∞)(Groeneboom, Jongbloed and Wellner (2001A, 2001B)). The iterative (2k− 1)-st spline algorithm was developed to compute the LSE of a k-monotone density on (0,∞ ...

format_quoteThe MLE and LSE of the Exponential density were computed for sample sizes n=100 and n=1000 with k=3 and k=6, revealing significant findings in the analysis.format_quote

Conjecture of error boundedness in a new Hermite interpolation problem via splines of odd-degree

by Fadoua Balabdaoui and Jon A. Wellner

We present a Hermite interpolation problem via splines of odd-degree which, to the best knowledge... more 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 interpolation via odd-degree splines. In this new interpolation problem, we conjecture that the interpolation error is bounded in the supremum norm independently of the locations of the knots. Given an integer k greater than

Estimation of a k-monotone density

by Fadoua Balabdaoui and Jon A. Wellner

characterizations, consistency and minimax lower bounds

Estimation of ak‐monotone density: characterizations, consistency and minimax lower bounds

by Fadoua Balabdaoui and Jon A. Wellner

Statistica Neerlandica, Dec 28, 2009

Densities with monotone or convex shape are encountered in many non-parametric estimation problem... more Densities with monotone or convex shape are encountered in many non-parametric estimation problems. Monotone densities arise naturally via connections with renewal theory and uniform mixing; see VARDI,(1989) and WOODROOFE and SUN (1993), for examples of the former, and WOODROOFE and SUN (1993), for the latter in an astronomical context. Estimation of monotone densities on (0,∞) was initiated by GRENANDER (1956a, b) with related work by AYER et al.(1955), BRUNk (1958), and VAN EEDEN (1957a, b). ...

Limit distribution theory for maximum likelihood estimation of a log-concave density

by Fadoua Balabdaoui and Jon A. Wellner

The Annals of Statistics, 2009

We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-c... more We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density, that is, a density of the form f 0 = exp ϕ 0 where ϕ 0 is a concave function on R. The pointwise limiting distributions depend on the second and third derivatives at 0 of H k , the "lower invelope" of an integrated Brownian motion process minus a drift term depending on the number of vanishing derivatives of ϕ 0 = log f 0 at the point of interest. We also establish the limiting distribution of the resulting estimator of the mode M(f 0 ) and establish a new local asymptotic minimax lower bound which shows the optimality of our mode estimator in terms of both rate of convergence and dependence of constants on population values.

A Kiefer-Wolfowitz theorem for convex densities

by Fadoua Balabdaoui and Jon A. Wellner

Institute of Mathematical Statistics Lecture Notes - Monograph Series, 2007

showed that if F is a strictly curved concave distribution function (corresponding to a strictly ... more showed that if F is a strictly curved concave distribution function (corresponding to a strictly monotone density f ), then the Maximum Likelihood Estimator Fn, which is, in fact, the least concave majorant of the empirical distribution function Fn, differs from the empirical distribution function in the uniform norm by no more than a constant times (n −1 log n) 2/3 almost surely. We review their result and give an updated version of their proof. We prove a comparable theorem for the class of distribution functions F with convex decreasing densities f , but with the maximum likelihood estimator Fn of F replaced by the least squares estimator Fn: if X 1 , . . . , Xn are sampled from a distribution function F with strictly convex density f , then the least squares estimator Fn of F and the empirical distribution function Fn differ in the uniform norm by no more than a constant times (n −1 log n) 3/5 almost surely. The proofs rely on bounds on the interpolation error for complete spline interpolation due to Hall [J. Approximation Theory 1 (1968) 209-218], Hall and Meyer [J. Approximation Theory 16 (1976) 105-122], building on earlier work by Birkhoff and de Boor [J. Math. Mech. 13 (1964) 827-835]. These results, which are crucial for the developments here, are all nicely summarized and exposited in de Boor [A Practical Guide to Splines (2001) Springer, New York].

On the Grenander estimator at zero

by Fadoua Balabdaoui and Jon A. Wellner

We establish limit theory for the Grenander estimator of a monotone density near zero. In particu... more 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 unbounded at zero, with different rates of growth to infinity. In the course of our study we develop new switching relations by use of tools from convex analysis. The theory is applied to a

format_quoteThe Grenander estimator is consistent on sets away from zero but fails at zero, demonstrating the complexity of estimating unbounded densities.format_quote

Limiting distribution of the MLE for current status data with competing risks

Page 1. GoBack Page 2. Limiting Distribution of the MLE for Current Status Data with Competing Ri... more Page 1. GoBack Page 2. Limiting Distribution of the MLE for Current Status Data with Competing Risks Swiss Statistics Seminar, November 16 2007, Luzern Marloes Maathuis ETH Zurich joint work with Piet Groeneboom and Jon Wellner Page 3. Motivating example: HIV/AIDS vaccine trials Current status data with competing risks - Marloes Maathuis (ETH Zurich) 2 / 39 • Promising candidate vaccines Page 4.

format_quoteNonparametric estimation relies on sub-distribution functions and their interrelations, critical for understanding overall failure distributions.format_quote