Christine H. Müller
Technical University of Dortmund
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Featured researches published by Christine H. Müller.
Journal of Statistical Planning and Inference | 1995
Christine H. Müller
Abstract For estimating a nonlinear aspect of a linear model maximin efficient designs are derived for the case that the support of the regarded designs is given and provides linearly independent regressors. Besides a general result two special results are presented, which provide maximin efficient designs under simple conditions. One of these results gives a simple condition so that the uniform design is maximin efficient. The other result deals with designs with a two-point support. These results have many applications. This is demonstrated by several examples including the problems of estimating the maximum point and the maximum value of quadratic response functions, the linear calibration problem, the problem of regression-based ratio estimation and the problems of estimating the relative effect of an additional factor and the equivalence of two treatments. Some of these examples repeat results which were already derived in the literature by straightforward methods. At last the application of the results to nonlinear models is shortly discussed.
Journal of Statistical Planning and Inference | 2003
Christine H. Müller; N. M. Neykov
Abstract Lower bounds for breakdown points of trimmed likelihood (TL) estimators in a general setup are expressed by the fullness parameter of Vandev (Statist. Probab. Lett. 16 (1993) 117) and results of Vandev and Neykov (Statistics 32 (1998) 111) are extended. A special application of the general result is the breakdown point behavior of TL estimators and related estimators as the S estimators in generalized linear models. For the generalized linear models, a connection between the fullness parameter and the quantity N (X) of Muller (J. Statist. Plann. Inference 45 (1995) 413) is derived for the case that the explanatory variables may not be in general position which happens in particular in designed experiments. These results are in particular applied to logistic regression and log-linear models where upper bounds for the breakdown points are also derived.
Journal of the American Statistical Association | 2004
Ivan Mizera; Christine H. Müller
This article introduces a halfspace depth in the location–scale model that is along the lines of the general theory given by Mizera, based on the idea of Rousseeuw and Hubert, and is complemented by a new likelihood-based principle for designing criterial functions. The most tractable version of the proposed depth—the Student depth—turns out to be nothing but the bivariate halfspace depth interpreted in the Poincaré plane model of the Lobachevski geometry. This fact implies many fortuitous theoretical and computational properties, in particular equivariance with respect to the Möbius group and favorable time complexities of algorithms. It also opens a way to introduce some other depth notions in the location–scale context, for instance, location–scale simplicial depth. A maximum depth estimator of location and scale—the Student median—is introduced. Possible applications of the proposed concepts are investigated on data examples.
Archive | 2003
N. M. Neykov; Christine H. Müller
A review of the studies concerning the finite sample breakdown point (BP) of the trimmed likelihood (TL) and related estimators based on the d—fullness technique of Vandev (1993), and Vandev and Neykov (1998) is made. In particular, the BP of these estimators in the frame of the generalized linear models (GLMs) depends on the trimming proportion and the quantity N(X) introduced by Muller (1995). A faster iterative algorithm based on resampling techniques for derivation of the TLE is developed. Examples of real and artificial data in the context of grouped logistic and log-linear regression models are used to illustrate the properties of the TLE.
Statistics & Probability Letters | 2002
Ivan Mizera; Christine H. Müller
The lower bounds for the explosion and implosion breakdown points of the simultaneous Cauchy M-estimator (Cauchy MLE) of the regression and scale parameters are derived. For appropriate tuning constants, the breakdown point attains the maximum possible value.
Journal of Statistical Planning and Inference | 1995
Christine H. Müller
Abstract For designed experiments we define the breakdown point of an estimator without allowing contaminated experimental conditions (explanatory, independent variables) because they are given by a fixed design. This provides a different definition of the breakdown point as it was used in former literature. For a wide class of estimators which we call h -trimmed weighted L p estimators and which includes high breakdown point estimators as the LMS estimator and the LTS estimators among others, we derive the breakdown point for situations which often appear in designed experiments. In particular, we derive the breakdown point for replicated experimental conditions and show that a design which maximizes the breakdown point should minimize the maximal number of experimental conditions which lie in a subspace of the parameter space. This provides a new optimality criterion for designs. This new optimality criterion leads to designs which are very different from the classical optimal designs. Two examples demonstrate the different behaviour.
Annals of Statistics | 2007
Martin Hillebrand; Christine H. Müller
The ability to remove a large amount of noise and the ability to preserve most structure are desirable properties of an image smoother. Unfortunately, they usually seem to be at odds with each other; one can only improve one property at the cost of the other. By combining M-smoothing and least-squares-trimming, the TM-smoother is introduced as a means to unify comer-preserving properties and outlier robustness. To identify edge-and comer-preserving properties, a new theory based on differential geometry is developed. Further, robustness concepts are transferred to image processing. In two examples, the TM-smoother outperforms other corner-preserving smoothers. A software package containing both the TM- and the M-smoother can be downloaded from the Internet.
Computational Statistics & Data Analysis | 2006
Tim Garlipp; Christine H. Müller
A two-step algorithm is proposed for estimating linear and circular shapes in noisy images. Initially and based on a previously proposed method, the pixels which are close to the edges of the shape are detected. These edges are assumed to be coming from a mixture of (linear or circular) regression functions and the parameters of these functions are estimated. An example with a triangle demonstrates the immense advantage of using an outlier robust estimator for the edge points. A second example deals with a problem from biology where the detection of circular shapes of fungi colonies is of interest.
Journal of Multivariate Analysis | 2009
Robin Wellmann; Peter Harmand; Christine H. Müller
A general approach for developing distribution-free tests for general linear models based on simplicial depth is presented. In most relevant cases, the test statistic is a degenerated U-statistic so that the spectral decomposition of the conditional expectation of the kernel function is needed to derive the asymptotic distribution. A general formula for this conditional expectation is derived. Then it is shown how this general formula can be specified for polynomial regression. Based on the specified form, the spectral decomposition and thus the asymptotic distribution is derived for polynomial regression of arbitrary degree. The power of the new test is compared via simulation with other tests. An application on cubic regression demonstrates the applicability of the new tests and in particular their outlier robustness.
Statistics | 1995
Christos P. Kitsos; Christine H. Müller
We regard the simple linear calibration problem where only the response y of the regression line y = β0 + β1 t is observed with errors. The experimental conditions t are observed without error. For the errors of the observations y we assume that there may be some gross errors providing outlying observations. This situation can be modeled by a conditionally contaminated regression model. In this model the classical calibration estimator based on the least squares estimator has an unbounded asymptotic bias. Therefore we introduce calibration estimators based on robust one-step-M-estimators which have a bounded asymptotic bias. For this class of estimators we discuss two problems: The optimal estimators and their corresponding optimal designs. We derive the locally optimal solutions and show that the maximin efficient designs for non-robust estimation and robust estimation coincide.