Lutz Duembgen

Universität Bern, Berna, Bern, Switzerland

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Publications (51)50.31 Total impact

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    Lutz Duembgen, Jon A. Wellner, Malcolm Wolff
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    ABSTRACT: 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 \begin{eqnarray*} \limsup_{n\rightarrow \infty} \left ( \frac{n}{2\log \log n} \right )^{1/3} ( \widehat{f}_n (t_0 ) - f(t_0) ) = \left| f(t_0) f'(t_0)/2 \right|^{1/3} 2M \end{eqnarray*} almost surely where $ M \equiv \sup_{g \in {\cal G}} T_g = (3/4)^{1/3}$ and $ T_g \equiv \mbox{argmax}_u \{ g(u) - u^2 \} $; here ${\cal G}$ 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.
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    Lutz Duembgen, Jon A. Wellner
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    ABSTRACT: We present a general law of the iterated logarithm for stochastic processes on the open unit interval having subexponential tails in a locally uniform fashion. It applies to standard Brownian bridge but also to suitably standardized empirical distribution functions. This leads to new goodness-of-fit tests and confidence bands which refine the procedures of Berk and Jones (1979) and Owen (1995). Roughly speaking, the high power and accuracy of the latter procedures in the tail regions of distributions are esentially preserved while gaining considerably in the central region.
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    ABSTRACT: We present new algorithms for $M$-estimators of multivariate location and scatter and for symmetrized $M$-estimators of multivariate scatter. The new algorithms are considerably faster than currently used fixed-point and other algorithms. The main idea is to utilize a Taylor expansion of second order of the target functional and devise a partial Newton-Raphson procedure. In connection with the symmetrized $M$-estimators we work with incomplete $U$-statistics to accelerate our procedures initially.
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    Lutz Duembgen, Markus Pauly, Thomas Schweizer
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    ABSTRACT: This survey provides a self-contained account of M-estimation of multivariate location and scatter, with special emphasis on maximum likelihood estimation for multivariate t-distributions. In particular, we present new proofs for existence of the underlying M-functionals, and discuss their weak continuity and differentiability. Moreover, we present M-estimation of scatter in a rather general framework with matrix-valued random variables. By doing so we reveal a connection between Tyler's (1987) M-functional of scatter and the estimation of proportional covariance matrices. Moreover, this general framework allows us to treat a new class of scatter estimators, based on symmetrizations of arbitrary order.
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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.
    Electronic Journal of Statistics 11/2013; 8. DOI:10.1214/14-EJS930 · 1.02 Impact Factor
  • Angelika Rohde, Lutz Dümbgen
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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 in Ann Stat 17:1001–1008, 1989). The present paper sheds new light on this gap. We develop exact and adaptive confidence regions 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 χ 2-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.
    Probability Theory and Related Fields 01/2013; 155(3-4). DOI:10.1007/s00440-012-0414-7 · 1.46 Impact Factor
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    ABSTRACT: Identifying and comparing different steady states is an important task for clinical decision making. Data from unequal sources, comprising diverse patient status information, have to be interpreted. In order to compare results an expressive representation is the key. In this contribution we suggest a criterion to calculate a context-sensitive value based on variance analysis and discuss its advantages and limitations referring to a clinical data example obtained during anesthesia. Different drug plasma target levels of the anesthetic propofol were preset to reach and maintain clinically desirable steady state conditions with target controlled infusion (TCI). At the same time systolic blood pressure was monitored, depth of anesthesia was recorded using the bispectral index (BIS) and propofol plasma concentrations were determined in venous blood samples. The presented analysis of variance (ANOVA) is used to quantify how accurately steady states can be monitored and compared using the three methods of measurement.
    E-Health and Bioengineering Conference (EHB), 2013; 01/2013
  • Society for Technology in Anesthesia - STA 2013 Annual Meeting, Phoenix, Arizona, USA; 01/2013
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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). DOI:10.1214/13-AOS1089 · 2.44 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 $\hat{P}$.
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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.
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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.
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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.
  • 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 03/2010; 80(5-6):376-380. DOI:10.1016/j.spl.2009.11.013 · 0.53 Impact Factor
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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; DOI:10.1214/10-AOS853 · 2.44 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 02/2010; 117(2):138-160. DOI:10.4169/000298910X476059 · 0.32 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; DOI:10.1524/stnd.2011.1073
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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) 06/2009; 71(3):549-592. DOI:10.1111/j.1467-9868.2009.00706.x · 5.72 Impact Factor
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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.
    SIAM Journal on Optimization 01/2009; 19(4). DOI:10.1137/050639132 · 2.11 Impact Factor

Publication Stats

764 Citations
50.31 Total Impact Points

Institutions

  • 2003–2013
    • Universität Bern
      • Institute of Mathematical Statistics and Actuarial Science
      Berna, Bern, Switzerland
  • 2011
    • Technische Universität Dresden
      Dresden, Saxony, Germany
  • 2009
    • University of Tampere
      Tammerfors, Province of Western Finland, Finland
  • 2008
    • University of Cambridge
      Cambridge, England, United Kingdom
  • 1996–2007
    • Stanford University
      Palo Alto, California, United States
  • 2006
    • Lomonosov Moscow State University
      Moskva, Moscow, Russia
  • 1998–2003
    • Universität zu Lübeck
      Lübeck Hansestadt, Schleswig-Holstein, Germany
  • 1994–1998
    • Universität Heidelberg
      • Institute of Applied Mathematics
      Heidelburg, Baden-Württemberg, Germany