Institut für Angewandte Mathematik und Statistik
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Publication Test for the model selection from two competing distribution classes(2016) Chen, Hong; Jensen, UweOne of the main tasks in statistics is to allocate an appropriate distribution function to a given set of data. Often the underlying distribution of the data can be approximated by a distribution function from a parametric distribution model class. This thesis deals with model selection from two given competing parametric model classes. To this end statistical hypothesis tests are proposed in different settings and their asymptotic behaviour for an increasing data size is analysed. This thesis is part of a DFG-project investigating the lifetime distribution of mechatronical systems such as DC-motors, which has been conducted in cooperation with engineers of the University of Stuttgart. The considered mechatronical systems are characterised by so-called covariates, which can influence the lifetime distribution. For DC-motors such covariates could be the electric current, the working load or the operation voltage. For instance, the lifetime distributions could be modelled by means of the Weibull distribution class or the log-normal distribution class with parameters depending linearly on the covariates. For a given data set an estimator for the unknown parameter in a model class can be obtained according to the maximum likelihood method. Under suitable conditions, the consistency of the estimator follows from the maximum likelihood theory for an increasing data size. In this thesis we consider two cases: First we handle the case with a fixed number of covariate values and the number of observations at each covariate value tending to infinity. After that, we consider the situation the other way round. The distance between the underlying distribution function and the competing model classes is defined based on the limit value of the maximum likelihood estimator and Cramér-von Mises distance. The reasons for the chosen distance measure are on the one hand the popularity of the maximum likelihood estimator and on the other hand the simple interpretability of the Cramér-von Mises distance with respect to our intention to approximate the lifetime distribution function. The null hypothesis is that both models provide an equally well fit. While the test statistic is defined by the estimated difference of the distances. Under suitable conditions, we show the asymptotic normality of the test statistic. Moreover, it is shown that the asymptotic variance can be estimated consistently by a plug-in estimator. With quantiles of the standard normal distribution for a given significance level the test decision rules are formulated. For the case with a fixed number of observations at each covariate and an increasing number of covariate values, the limit of the maximum likelihood estimator is defined analogously. The distance is adjusted accordingly and in the test statistic the empirical distribution is replaced by the Nadaraya-Watson kernel estimator. For one dimensional covariates we show similar results as in the first case. However, it cannot be extended to the multidimensional case in general. Thus, a one-sided test is proposed. Further, the consistency of the test is also proven. The results are extended to the case with right random censoring, whereby the Kaplan-Meier and the Beran estimator for distribution functions are used. At the end of the thesis the applicability of the proposed hypothesis tests is evaluated by means of simulations and a case study.Publication SIA matrices and non-negative stationary subdivision(2012) Li, Xianjun; Jetter, KurtThis dissertation is concerned with SIA matrices and non-negative stationary subdivision, and is organized as follows: After an introducing chapter where some basic notation is given we describe, in Chapter 3, how non-negative subdivision is connected to a corresponding non-homogenous Markov process. The family of matrices A, built from the mask of the subdivision scheme, is introduced. Among other results, Lemma 3.1 and Lemma 3.2 relate the coefficients of the iterated masks to matrix products from the family A, and in the limiting case the values of the basic limit function are found from the entries in an infinite product of matrices. Chapter 4 and Chapter 5 are the core of this dissertation. In Chapter 4, we first review some spectral and graph properties of row-stochastic matrices and, in particular, of SIA matrices. We point to the notion of scrambling power, introduced by Hajnal [16], and of the related coefficient of ergodicity. We also consider the directed graph of such matrices, and we improve upon a condition given by Ren and Beard in [30]. Then we study finite families of SIA matrices, the properties of their indicator matrices and the connectivity of their directed graphs. We consider this chapter to be an important contribution to the theory of non-negative subdivision, since it explains the background in order to apply the convergence result of Anthonisse and Tijms [2], which we reprove in Section 4.6, to rank one convergence of infinite products of row stochastic matrices. It does not use the notion of joint spectral radius but the (equivalent) coefficient of ergodicity. Properties equivalent to SIA are listed in Lemma 4.7 and in the subsequent Lemma 4.8; they connect the SIA property to equivalent conditions (scrambling property, positive column property) as they appear in the existing literature dealing with convergence of non-negative subdivision. The fifth chapter of the dissertation contains the full proof of the characterization of uniform convergence for non-negative subdivision, for the univariate and bivariate case, the latter one being a representative for multivariate aspects. It uses the pointwise definition of the limit function at dyadic points - refering to the dyadic expansion of real vectors from the unit cube - using the Anthonisse-Tijms pointwise convergence result, and employs the proper extension of the Micchelli-Prautzsch compatibility condition to the multivariate case, taking care of the ambiguity of representation of dyadic points. As a consequence, the Hölder exponent of the basic limit function can be expressed in terms of the coefficient of ergodicity of the family A. Our convergence theorems, in Theorem 5.1 and Theorem 5.8, include the existing characterizations of uniform convergence for non-negative univariate and bivariate subdivision from the literature, except for the GCD condition, which seems to be a condition applicable to univariate subdivision only. Chapter 5 also reports on some further attempts where we have tried to extend conditions from univariate subdivision, which are sufficient for convergence, to the bivariate case. We could find a bivariate analogue of Melkman's univariate string condition, which we call - in the bivariate case - a rectangular string condition. The chapter concludes with stating the fact that uniform convergence of non-negative stationary subdivision is a property of the support of the mask alone, modulo some apparent necessary conditions such as the sum rules. A typical application of this support property characterizes uniform convergence in the case where the mask is a convex combination of other masks. The dissertation ends with two short chapters on tensor product and box spline subdivision, and an appendix where some definitions and useful lemmas and theorems about matrix and graph theory are stated without proofs.Publication Cox-Type regression and transformation models with change-points based on covariate thresholds(2007) Lütkebohmert-Marhenke, Constanze; Jensen, UweIn this thesis we consider Cox-type regression models and transformation models for right-censored survival time data with bent-line change-points in the underlying regression functions according to covariate thresholds. We establish the usual asymptotic properties of the estimates such as √(n) consistency and asymptotic normality. Furthermore, we applied the Cox regression model with change-points to different data sets.