Lutz Duembgen

Technische Universität Dresden, Dresden, Saxony, Germany

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Publications (38)46.9 Total impact

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    ABSTRACT: We consider nonparametric maximum-likelihood estimation of a log-concave density in case of interval- or right-censored or binned data. Theoretical properties are studied and an algorithm is proposed for the approximate computation of the estimator.
    11/2013;
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    ABSTRACT: Many preschool children have wheeze or cough, but only some have asthma later. Existing prediction tools are difficult to apply in clinical practice or exhibit methodological weaknesses. We sought to develop a simple and robust tool for predicting asthma at school age in preschool children with wheeze or cough. From a population-based cohort in Leicestershire, United Kingdom, we included 1- to 3-year-old subjects seeing a doctor for wheeze or cough and assessed the prevalence of asthma 5 years later. We considered only noninvasive predictors that are easy to assess in primary care: demographic and perinatal data, eczema, upper and lower respiratory tract symptoms, and family history of atopy. We developed a model using logistic regression, avoided overfitting with the least absolute shrinkage and selection operator penalty, and then simplified it to a practical tool. We performed internal validation and assessed its predictive performance using the scaled Brier score and the area under the receiver operating characteristic curve. Of 1226 symptomatic children with follow-up information, 345 (28%) had asthma 5 years later. The tool consists of 10 predictors yielding a total score between 0 and 15: sex, age, wheeze without colds, wheeze frequency, activity disturbance, shortness of breath, exercise-related and aeroallergen-related wheeze/cough, eczema, and parental history of asthma/bronchitis. The scaled Brier scores for the internally validated model and tool were 0.20 and 0.16, and the areas under the receiver operating characteristic curves were 0.76 and 0.74, respectively. This tool represents a simple, low-cost, and noninvasive method to predict the risk of later asthma in symptomatic preschool children, which is ready to be tested in other populations.
    The Journal of allergy and clinical immunology 07/2013; · 12.05 Impact Factor
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    ABSTRACT: We derive multiscale statistics for deconvolution in order to detect qualitative features of the unknown density. An important example covered within this framework is to test for local monotonicity on all scales simultaneously. We investigate the moderately ill-posed setting, where the Fourier transform of the error density in the deconvolution model is of polynomial decay. For multiscale testing, we consider a calibration, motivated by the modulus of continuity of Brownian motion. We investigate the performance of our results from both the theoretical and simulation based point of view. A major consequence of our work is that the detection of qualitative features of a density in a deconvolution problem is a doable task although the minimax rates for pointwise estimation are very slow.
    The Annals of Statistics 07/2011; 41(3). · 2.53 Impact Factor
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    Lutz Duembgen, Perla Zerial
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    ABSTRACT: Let $P$ be a probability distribution on $q$-dimensional space. The so-called Diaconis-Freedman effect means that for a fixed dimension $d << q$, most $d$-dimensional projections of $P$ look like a scale mixture of spherically symmetric Gaussian distributions. The present paper provides necessary and sufficient conditions for this phenomenon in a suitable asymptotic framework with increasing dimension $q$. It turns out, that the conditions formulated by Diaconis and Freedman (1984) are not only sufficient but necessary as well. Moreover, letting $\hat{P}$ be the empirical distribution of $n$ independent random vectors with distribution $P$, we investigate the behavior of the empirical process $\sqrt{n}(\hat{P} - P)$ under random projections, conditional on $\hat{P}$.
    07/2011;
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    ABSTRACT: This paper introduces and analyzes a stochastic search method for parameter estimation in linear regression models in the spirit of Beran and Millar (1987). The idea is to generate a random finite subset of a parameter space which will automatically contain points which are very close to an unknown true parameter. The motivation for this procedure comes from recent work of Duembgen, Samworth and Schuhmacher (2011) on regression models with log-concave error distributions.
    06/2011;
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    ABSTRACT: We derive multiscale statistics for deconvolution in order to detect qualitative features of the unknown density. An important example covered within this framework is to test for local monotonicity on all scales simultaneously. The errors in the deconvolution model are restricted to a certain class of distributions that include Laplace, Gamma and Exponential random variables. Our approach relies on inversion formulas for deconvolution operators. For multiscale testing, we consider a calibration, motivated by the modulus of continuity of Brownian motion. We investigate the performance of our results from both the theoretical and simulation based point of view. A major consequence of our work is that the detection of qualitative features of a density in a deconvolution problem is a doable task although the minimax rates for pointwise estimation are very slow.
    01/2011;
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    Lutz Duembgen
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    ABSTRACT: We review various inequalities for Mills' ratio (1 - \Phi)/\phi, where \phi and \Phi denote the standard Gaussian density and distribution function, respectively. Elementary considerations involving finite continued fractions lead to a general approximation scheme which implies and refines several known bounds.
    12/2010;
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    ABSTRACT: We study the approximation of arbitrary distributions $P$ on $d$-dimensional space by distributions with log-concave density. Approximation means minimizing a Kullback--Leibler-type functional. We show that such an approximation exists if and only if $P$ has finite first moments and is not supported by some hyperplane. Furthermore we show that this approximation depends continuously on $P$ with respect to Mallows distance $D_1(\cdot,\cdot)$. This result implies consistency of the maximum likelihood estimator of a log-concave density under fairly general conditions. It also allows us to prove existence and consistency of estimators in regression models with a response $Y=\mu(X)+\epsilon$, where $X$ and $\epsilon$ are independent, $\mu(\cdot)$ belongs to a certain class of regression functions while $\epsilon$ is a random error with log-concave density and mean zero.
    The Annals of Statistics 02/2010; · 2.53 Impact Factor
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    ABSTRACT: 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 $(\B,\|\cdot\|)$ there exists a constant $K = K(\B,\|\cdot\|)$ such that for arbitrary independent and centered random vectors $X_1, X_2, ..., X_n \in \B$, their sum $S_n$ satisfies the inequality $ E \|S_n \|^2 \le K \sum_{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.
    The American Mathematical Monthly 01/2010; 117(2):138-160. · 0.29 Impact Factor
  • Dominic Schuhmacher, Lutz Dümbgen
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    ABSTRACT: This note proves Hellinger-consistency for the non-parametric maximum likelihood estimator of a log-concave probability density on .
    Statistics [?] Probability Letters 01/2010; 80(5-6):376-380. · 0.53 Impact Factor
  • Lutz Dümbgen
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    ABSTRACT: In diesem Kapitel betrachten wir ein Variablenpaar (X,Y) bestehend aus einer “unabhängigen Variable” X mit beliebigem Wertebereich X und einer “abhängigen Variable” oder “Response” Y ∈ ℝ. Die Frage ist, inwiefern die Response Y von X abhängt. Typischerweise ist X ein Vektor von diversen Variablen.
    12/2009: pages 187-223;
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    ABSTRACT: In this paper we show that the family P_d of probability distributions on R^d with log-concave densities satisfies a strong continuity condition. In particular, it turns out that weak convergence within this family entails (i) convergence in total variation distance, (ii) convergence of arbitrary moments, and (iii) pointwise convergence of Laplace transforms. Hence the nonparametric model P_d has similar properties as parametric models such as, for instance, the family of all d-variate Gaussian distributions.
    07/2009;
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    ABSTRACT: The computation of robust regression estimates often relies on minimization of a convex functional on a convex set. In this paper we discuss a general technique for a large class of convex functionals to compute the minimizers iteratively which is closely related to majorization-minimization algorithms. Our approach is based on a quadratic approximation of the functional to be minimized and includes the iteratively reweighted least squares algorithm as a special case. We prove convergence on convex function spaces for general coercive and convex functionals F and derive geometric convergence in certain unconstrained settings. The algorithm is applied to TV penalized quantile regression and is compared with a step size corrected Newton-Raphson algorithm. It is found that typically in the first steps the iteratively reweighted least squares algorithm performs significantly better, whereas the Newton type method outpaces the former only after many iterations. Finally, in the setting of bivariate regression with unimodality constraints we illustrate how this algorithm allows to utilize highly efficient algorithms for special quadratic programs in more complex settings.
    01/2009;
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    ABSTRACT: A general method for exploring multivariate data by comparing different estimates of multivariate scatter is presented. The method is based on the eigenvalue-eigenvector decomposition of one scatter matrix relative to another. In particular, it is shown that the eigenvectors can be used to generate an affine invariant co-ordinate system for the multivariate data. Consequently, we view this method as a method for "invariant co-ordinate selection". By plotting the data with respect to this new invariant co-ordinate system, various data structures can be revealed. For example, under certain independent components models, it is shown that the invariant co- ordinates correspond to the independent components. Another example pertains to mixtures of elliptical distributions. In this case, it is shown that a subset of the invariant co-ordinates corresponds to Fisher's linear discriminant subspace, even though the class identifications of the data points are unknown. Some illustrative examples are given. Copyright (c) 2009 Royal Statistical Society.
    Journal of the Royal Statistical Society Series B (Statistical Methodology) 01/2009; 71(3):549-592. · 4.81 Impact Factor
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    Rudolf Beran, Lutz Duembgen
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    ABSTRACT: In this paper we describe active set type algorithms for minimization of a smooth function under general order constraints, an important case being functions on the set of bimonotone r-by-s matrices. These algorithms can be used, for instance, to estimate a bimonotone regression function via least squares or (a smooth approximation of) least absolute deviations. Another application is shrinkage estimation in image denoising or, more generally, regression problems with two ordinal factors after representing the data in a suitable basis which is indexed by pairs (i,j) in {1,...,r}x{1,...,s}. Various numerical examples illustrate our methods.
    Statistics and Computing 09/2008; · 1.98 Impact Factor
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    Madeleine L. Cule, Lutz Duembgen
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    ABSTRACT: In this note we provide explicit expressions and expansions for a special function which appears in nonparametric estimation of log-densities. This function returns the integral of a log-linear function on a simplex of arbitrary dimension. In particular it is used in the R-package "LogCondDEAD" by Cule et al. (2007).
    08/2008;
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    Lutz Duembgen, Arne Kovac
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    ABSTRACT: Suppose that we observe independent random pairs (X_1,Y_1), (X_2,Y_2), ..., (X_n,Y_n).Our goal is to estimate regression functions such as the conditional mean or beta-quantile of Y given X, where 0 < beta < 1. In order to achieve this we minimize criteria such as, for instance, the sum of rho(f(X_i) - Y_i) over all i plus lambda * TV(f) among all candidate functions. Here rho(.) is some convex function depending on the particular regression function we have in mind, TV(f) stands for the total variation of f, and lambda > 0 is some tuning parameter. This framework is extended further to include binary or Poisson regression, and to include localized total variation penalties. The latter are needed to construct estimators adapting to inhomogeneous smoothness of f. For the general framework we develop noniterative algorithms for the solution of the minimization problems which are closely related to the taut string algorithm (cf. Davies and Kovac, 2001). Further we establish a connection between the present setting and monotone regression, extending previous work by Mammen and van de Geer (1997).
    Electronic Journal of Statistics 04/2008; · 0.79 Impact Factor
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    Angelika Rohde, Lutz Duembgen
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    ABSTRACT: In the setting of high-dimensional linear models with Gaussian noise, we investigate the possibility of confidence statements connected to model selection. Although there exist numerous procedures for adaptive point estimation, the construction of adaptive confidence regions is severely limited (cf. Li, 1989). The present paper sheds new light on this gap. We develop exact and adaptive confidence sets for the best approximating model in terms of risk. One of our constructions is based on a multiscale procedure and a particular coupling argument. Utilizing exponential inequalities for noncentral chi-squared distributions, we show that the risk and quadratic loss of all models within our confidence region are uniformly bounded by the minimal risk times a factor close to one.
    02/2008;
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    Lutz Duembgen, Bernd-Wolfgang Igl, Axel Munk
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    ABSTRACT: Let $(X,Y)$ be a random variable consisting of an observed feature vector $X\in \mathcal{X}$ and an unobserved class label $Y\in \{1,2,...,L\}$ with unknown joint distribution. In addition, let $\mathcal{D}$ be a training data set consisting of $n$ completely observed independent copies of $(X,Y)$. Usual classification procedures provide point predictors (classifiers) $\widehat{Y}(X,\mathcal{D})$ of $Y$ or estimate the conditional distribution of $Y$ given $X$. In order to quantify the certainty of classifying $X$ we propose to construct for each $\theta =1,2,...,L$ a p-value $\pi_{\theta}(X,\mathcal{D})$ for the null hypothesis that $Y=\theta$, treating $Y$ temporarily as a fixed parameter. In other words, the point predictor $\widehat{Y}(X,\mathcal{D})$ is replaced with a prediction region for $Y$ with a certain confidence. We argue that (i) this approach is advantageous over traditional approaches and (ii) any reasonable classifier can be modified to yield nonparametric p-values. We discuss issues such as optimality, single use and multiple use validity, as well as computational and graphical aspects.
    Electronic Journal of Statistics 02/2008; · 0.79 Impact Factor
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    Lutz Duembgen, Kaspar Rufibach
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    ABSTRACT: We study nonparametric maximum likelihood estimation of a log--concave probability density and its distribution and hazard function. Some general properties of these estimators are derived from two characterizations. It is shown that the rate of convergence with respect to supremum norm on a compact interval for the density and hazard rate estimator is at least (log(n)/n)^{1/3} and typically (log(n)/n)^{2/5} whereas the difference between the empirical and estimated distribution function vanishes with rate o_p (n^{-1/2}) under certain regularity assumptions.
    Bernoulli 10/2007; · 0.94 Impact Factor

Publication Stats

329 Citations
46.90 Total Impact Points

Institutions

  • 2011
    • Technische Universität Dresden
      Dresden, Saxony, Germany
  • 2003–2010
    • Universität Bern
      • Institut für mathematische Statistik und Versicherungslehre
      Bern, BE, Switzerland
    • University of Rostock
      • Institut für Anatomie
      Rostock, Mecklenburg-Vorpommern, Germany
  • 2009
    • University of Tampere
      Tammerfors, Province of Western Finland, Finland
  • 2008
    • University of Cambridge
      Cambridge, England, United Kingdom
  • 2007
    • Stanford University
      Palo Alto, California, United States
  • 2006
    • Lomonosov Moscow State University
      Moskva, Moscow, Russia
  • 1998
    • Universität zu Lübeck
      Lübeck Hansestadt, Schleswig-Holstein, Germany
    • Universität Heidelberg
      • Institute of Applied Mathematics
      Heidelburg, Baden-Württemberg, Germany