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Dissertation, RWTH Aachen University, 2021; Aachen : RWTH Aachen University 1 Online-Ressource : Illustrationen, Diagramme (2021). = Dissertation, RWTH Aachen University, 2021, In this thesis, we discussed (simple) SSALT with two stress factors for exponentially distributed lifetimes. Under the common CE-assumption and stating a log-linear life-stress relationship, we considered the role of step-stress testing in ALT in general and different specific aspects of bivariate (simple) SSALT. We proved that for a wide range of test designs for two stress factors applying arbitrary time varying or constant stress functions a statistically equivalent bivariate simple step-stress test can be constructed, resulting in the same asymptotic variance matrix as the initial test design. Therefore, if the objective of the design procedure is to optimize a criterion solely based on the Fisher information matrix as it is common in practice and research, the consideration of simple step-stress plans is sufficient. In other words, the motivation of alternative stress loadings like k-level constant stress, ramp-stress, or multiple step-stress designs requires additional alternative objectives that do not depend on the Fisher information matrix like the minimization of the probability of non-existent estimates or achieving desirable properties for small sample sizes. For given stress levels of a bivariate simple step-stress test, we were able to find the optimal change points minimizing the asymptotic variance of the logarithm of the MLE of the mean time to failure under NOC for step-up and step-down tests. The fact that the optimal change points were derived in closed form enables the derivation of a lower bound for the asymptotic variance of bivariate simple step-stress tests in general, only depending on the larger amount of extrapolation between the two stress factors. Further results on optimal test designs were deduced also considering the minimization of the non-existence probability as a secondary criterion when the asymptotic variance is fixed to a given level. In order to account for the influence of misspecified necessary pre-estimates of the model parameters on the estimation based on a resulting non-optimal test plan, the change points were derived under these circumstances. We found a representation of the asymptotic variance of the MLE under a non-optimal test design as a function of the relative amounts of misspecification. This allows to calculate the caused deviation from the minimal possible asymptotic variance based on the choice of stress levels and ranges of misspecification alone without any knowledge of the model parameters. As another important aspect in the conduction of SSALT experiments, we compared different methods to estimate the parameters of the link function between the lifetime distribution and the stress factors for type-II censored data. We proposed a least squares approach based on the MLEs of the scale parameters on the respective increased stress levels which always leads to closed form estimates and outlined prerequisites for equivalence to the standard direct ML estimation of the log-link parameters. In a situation where the two methods do not coincide, we proved the existence of the MLEs of the link function parameters and found conditions that ensure receiving the estimators in closed form as well. Furthermore, alternative consistent closed form estimators of the log-link parameters based on the MLEs of the scale parameters were motivated. The extension of known results on the distribution of these MLEs to multiple step-stress situations allowed to deduce exact distributional properties of the MLE of the mean lifetime under NOC., Published by RWTH Aachen University, Aachen

Dissertation, RWTH Aachen University, 2021; Aachen : RWTH Aachen University 1 Online-Ressource : Illustrationen (2021). = Dissertation, RWTH Aachen University, 2021, To analyze the fairness of an educational system of a country and to help with development of pedagogical concepts, questionnaire and test based surveys are important tools. An essential challenge in conducting such surveys is the measurement of not directly observable traits such as the ability of students in different subjects. These traits are modeled by latent variables. This thesis restricts on dichotomous items where the possible responses to each item can be categorized in a set of two options (e.g., "correct" and "incorrect") and on continuous latent variables. In item response theory (IRT) the probability of a correct response to an item depending on the latent variable is modeled. Multidimensional models suppose that there are several latent variables which are collected in a latent vector. Chapter 1 provides an overview of IRT models and methods for estimating model parameters and latent vectors. A particular emphasis lies on generalized linear latent variable models (GLLVM) and models that have a closed form expression of the marginal distribution of the response vector. Chapter 2 introduces an extension of GLLVM with respect to link functions and distributions of the latent vector that depend on parameters for their respective shapes. It is pointed out how this is connected to several models in the literature which are unified in this class. The consistency and asymptotic efficiency of the marginal maximum likelihood estimator (MMLE) for the model parameters is proved. This also implies that these asymptotic properties hold for many classic models, thus contributing to the estimation theory for IRT models in general. The asymptotic chi-square distribution of Wald, score and likelihood-ratio test-statistics is derived using the asymptotic efficiency of the MMLE. Model fitting, estimation of latent traits, nested model tests and model selection are studied in simulation studies. In Chapter 3 the asymptotic theory of estimating latent vectors is discussed. The estimation of latent vectors can be interpreted as (empirical) Bayesian point estimation with previous estimation of the (multidimensional) IRT model parameters. A primary target of this chapter is the investigation of variants of a Bernstein-von Mises theorem of latent vectors, i.e. the asymptotic posterior normality (APN) of latent vectors. This chapter provides a comprehensive analysis of questions related to Bernstein-von Mises theorems and the asymptotics of latent vector estimation for binary IRT. Current results regarding the asymptotics of the posterior of a single latent variable in the IRT literature are extended with respect to the multivariate case but also to the type of the convergence, the considered estimators and their asymptotic efficiency. In Chapter 4 a linear approximation of the expected a-posteriori estimator (aEAP) for latent vectors is obtained using the component statistics and the APN theory of Chapter 3. Properties of the aEAP are examined using a simulation study. A new EM-algorithm for MMLE of high dimensional logit models is derived using the APN theory once more and combining it with the aEAP. This EM-algorithm is easy to implement for any dimension of the latent vector by simplifying steps of similar adaptive algorithms for high dimensional settings. Chapter 5 focuses on parameter estimation for large high-dimensional IRT settings in which classic methods are unfeasible. Based on a pseudo likelihood procedure for a class of generalized IRT models that cannot always be interpreted as latent variable models, a method is obtained whose resulting fitted models are guaranteed to be equivalent to latent variable models. The implemented procedure is fast but the parameter estimates are biased. Bias and efficiency of the estimator are studied via simulations., Published by RWTH Aachen University, Aachen

Dissertation, RWTH Aachen University, 2020; Aachen 1 Online-Ressource (vi, 165 Seiten) : Illustrationen, Diagramme (2020). = Dissertation, RWTH Aachen University, 2020, In reliability theory and applications, modeling the lifetimes of technical systems with several components plays an important role. For instance, interest may lie in describing the lifetimes of components within k-out-of-n systems which consist of n identical components and work as long as at least k of these components are running. In many such systems, the remaining components experience an increased load after some component has failed. This effect, also called load-sharing effect, can be modeled, e.g., by sequential order statistics which were introduced as a generalization of common order statistics. In this thesis, a sub-model ensuring proportional hazard rates is examined for the case of an underlying exponential or Weibull distribution. Here, the focus lies on questions regarding the fitting of this model to given data. More detailed, topics of classical statistical inference such as point estimation, hypothesis testing, and confidence sets are discussed. To this end, the underlying structure of transformation models proves helpful which gain much attention in the literature next to the theory of exponential families. For the model of sequential order statistics with an underlying exponential distribution, the transformation model approach leads to the minimum risk equivariant estimators of the model parameters as an alternative to the known maximum likelihood estimator (MLE) or the uniformly minimum variance unbiased estimator. In addition, we present a method to derive exact goodness-of-fit tests on this model. For the model with an underlying Weibull distribution, we start by proving certain regularity conditions which lead to the Fisher information matrix and which, eventually, implicate the consistency and asymptotic efficiency of the MLE of the model parameters. Moreover, the MLE is seen to satisfy certain pivotal properties where the distributions of several quantities comprising the estimator and the parameters are independent of the true underlying parameters. These properties are, in fact, shown to be a consequence of the equivariance of the MLE, which also proves them for a much larger class of estimators. We demonstrate that the MLE of the shape parameter of the underlying Weibull distribution is biased and subsequently discuss several methods for reducing this bias, leading to a variety of other equivariant estimators. By means of simulation and by utilizing the pivotal properties mentioned earlier, these alternative estimators are shown to be superior in terms of variance and mean squared error as well. Different null hypotheses, e.g. for testing the adequacy of one particular model or the presence of a load-sharing effect, are discussed. Here, three well-known test statistics given by the likelihood ratio, Rao’s score, and Wald’s statistic are applied which usually lead to asymptotic tests. However, the transformation model structure allows for the derivation of exact tests based on these statistics which can also be compared much easier via simulations. Thereafter, exact and asymptotic confidence sets for the Weibull shape parameter and for the model parameters of the sequential order statistics are addressed. In the case where the model parameters are known and the Weibull shape parameter is the only unknown parameter, the resulting univariate log-likelihood function may have multiple local maxima which might lead to problems when trying to find the MLE. This circumstance is further analyzed and a possible solution is addressed. We observe that, other than in a similar and well-known situation concerning samples from a Cauchy distribution with an unknown location parameter, this problem is seen to be caused by the corresponding Kullback-Leibler divergence having multiple local minima. Finally, two approaches are proposed to generalize the model of sequential order statistics from an underlying Weibull distribution to other distributions that stem from a log-location-scale family of distributions. Here, several properties are seen to be maintained during this transition depending on whether the transformation model structure or a proportional hazard rate property are conserved. For illustration, the methods derived in this thesis are applied to two real data sets discussed in the literature., Published by Aachen

Dissertation, RWTH Aachen University, 2018; Aachen 1 Online-Ressource (v, 214 Seiten) : Illustrationen (2018). = Dissertation, RWTH Aachen University, 2018, Due to recent advancements in fields such as information technology and genomics, nowadays one commonly faces high-dimensional data where the number of explanatory variables is possibly much larger than the number of observations. In such situations one is particularly interested in variable selection, meaning that one aims at identifying a sparse model with a relatively small subset of variables that fits and ideally explains the observed data well. This thesis deals with the variable selection problem in the setting of high-dimensional generalized linear models (GLMs). While many variable selection methods like the Lasso are based on solving convex $\ell_1$-type relaxations of the original problem, a main motive of this work is the desire to provide solutions to generally NP-hard $\ell_0$-regularized problems induced by model selection criteria such as the Extended Bayesian Information Criterion (EBIC). For this purpose, the Adaptive Subspace (AdaSub) method is proposed which is based on the idea of adaptively solving several low-dimensional sub-problems of the original high-dimensional problem. AdaSub is a stochastic algorithm which sequentially adapts the sampling probabilities of the individual variables based on their currently estimated "importance". It is shown that the updating scheme of AdaSub can be motivated in a Bayesian way and that the method "converges correctly" against the best model according to the employed criterion, provided that the so-called ordered importance property (OIP) is satisfied. Furthermore, the variable selection consistency of AdaSub is proved under suitable conditions. Since solving the sampled sub-problems can be computationally expensive for GLMs different than the normal linear model, "greedy" modifications of AdaSub are introduced which provide approximate solutions to the sub-problems. It is argued that BackAdaSub, a version of AdaSub based on Backward Stepwise Selection, may be used as an efficient surrogate algorithm. The "correct convergence" of BackAdaSub can be guaranteed under the modified ordered importance property (MOIP), which is a stronger condition than the original OIP. The performance of AdaSub and BackAdaSub in comparison to other prominent competitors such as the Lasso, the Adaptive Lasso, the SCAD and Stability Selection is investigated via various simulated and real data examples in the framework of linear and logistic regression models. Finally, a Metropolized version of AdaSub, called the MAdaSub algorithm, is proposed for sampling from posterior model distributions in the Bayesian variable selection context. MAdaSub is an adaptive Markov Chain Monte Carlo (MCMC) algorithm which sequentially adjusts the proposal distribution based on the information from the previously sampled models. It is shown that the MAdaSub algorithm is ergodic despite its continuing adaptation, i.e. "in the limit" it samples from the correct target distribution. Through simulated and real data examples it is demonstrated that MAdaSub can provide stable estimates of posterior marginal inclusion probabilities even for very high-dimensional and multimodal posterior model distributions., Published by Aachen

Dissertation, RWTH Aachen University, 2018; Aachen 1 Online-Ressource (iv, 178 Seiten) : Illustrationen (2018). = Dissertation, RWTH Aachen University, 2018, In this doctoral thesis, we establish a method which aims to improve the maximum likelihood estimator (MLE) in situations, when a pre-estimate of the underlying parameter of a probability distribution is available. Especially when the sample size is small, the MLE may have high variance and may lead to poor estimates. The main idea is to incorporate the pre-estimate to settle this drawback. This procedure is an alternative approach to the classical Bayesian paradigm in order to construct an estimator which contains prior knowledge. The construction is based on a minimum divergence approach. Given the set of all estimates having the same distance to both, the ML-estimate and the pre-estimate, we are looking fora parameter which approximates best the ML estimate and the pre-estimate. The resulting estimator is called the Equal Distance Estimator (EDE). Basically, such procedure defines a shrinkage estimator, since geometrically viewed, any ML-estimate is dragged closer to the pre-estimate. Explicit forms of the EDE are given for different divergences and a wide class of probability distributions. Moreover, conditions for existence and uniqueness of the EDEare established. Many properties of the MLE are bequeathed to the EDE. The most important one is invariance w.r.t. parametrizations. Given multinomial data, a slight extension of the EDE leads to a new approach to deal with sparsity. As a performance criterion we use Pitman’s measure of closeness to compare the EDE with the MLE. Finally we introduce an update procedure of the EDE for new samples from a multivariate normal distribution., Published by Aachen

RWTH Aachen University, Diss., 2017; 221 pp. (2017). = RWTH Aachen University, Diss., 2017, This Ph.D. thesis deals with one of the fundamental problems of categorical data analysis, namely that of measuring the association between categorical variables, cross-classified in a two-way table. Such tables occur in many scientific fields such as economics, social and biomedical sciences. Although a sensitive and more informative analysis is provided through adequate models, which constitute a basic and flexible tool, their implementation and interpretation often require advanced model fitting procedures and statistical software skills that can be too complex for practitioners. Association measures provide a convenient alternative offering a compact identification and overall quantification of underlying association. They are easy to understand and interpret. This thesis develops new association measures for 2 x 2 tables based on the phi-divergence by generalising the most fundamental measure of association, the odds ratio. The adopted approach is motivated by an extensive study on continuity corrections and confidence interval construction techniques, which are approaches for dealing with the problems caused by sampling zeros, i.e. cells with observed zero frequencies. Sampling zeros may lead to infinite estimates of the log-odds ratio and prohibit the use of asymptotic inferential methods due to infinite asymptotic variance estimates. The newly introduced measure, the phiscaled odds ratio, aims at solving these deficiencies by using a phi-divergence induced scale change. A scale change can improve the compatibility with sampling zeros and can -- in some set-ups -- lead to better Wald confidence intervals for the phi-scaled odds ratios with respect to their coverage probability and average relative length. A scalar measure can often be misleading in I x J tables when the association structure is more complex and cannot be described by a single parameter. The classical generalised odds ratios are naturally linked to parameters of association models. This close connection is used to construct new non-scalar measures of association. These measures are more informative since they inherit the increased sensibility of models and offer more options to cover association structures without losing the easy interpretability. Closed-form estimators for these model-based measures are introduced which are close to the maximum likelihood estimators, which have to be computed iteratively. A scale change can lead to more adequate measures. Therefore, this model-based approach is extended using the phi-divergence by providing and studying new generalised phi-scaled odds ratios for I x J tables. They are linked to a new phi-scaled association model, the generalised phi-linear model, and thus provide a phi-scaled extension of the modelbased measures for which closed-form estimators are also developed. I x I square tables with commensurable classification variables are of special interest, e.g. in social mobility studies to value the permeability of economical systems. Such tables can be analysed with symmetry models. The already existing phi-scaled symmetry models form the basis to develop a phiscaled asymmetry measure. Thus, a new family of directed asymmetry measures is introduced along with new phi-scaled versions of the standard symmetry tests of McNemar and Bowker. The main contribution of this work is the exploration and signalisation of the great flexibility of phi-divergence based categorical data measures, thus paving the way for further research, among others, on small-sized multi-way tables, which are naturally confronted with the presence of sampling zeros., Published by Aachen

RWTH Aachen University, Diss., 2017; Aachen, 1 Online-Ressource (ix, 161 Seiten) : Illustrationen, Diagramme(2017). = RWTH Aachen University, Diss., 2017, The two classical approaches in estimation theory are point estimation and confidence interval estimation. A confidence interval contains the unknown parameter of interest of a parametric family of distributions with a probability greater than or equal to a certain value, called confidence coefficient or confidence level. If we deal with multiparametric families of distributions, we are interested in simultaneously estimating the parameter vector by determining a multidimensional confidence region. In doing so, several difficulties may occur. One of the difficulties arising is that of finding a suitable pivot statistic for the parameter vector. A pivot statistic is a statistic whose distribution does not depend on the parameter itself. Another difficulty arises from the fact that we usually want to predetermine the confidence coefficient of the confidence region. In most of the (standard) methods developed we need to allocate a certain confidence level to every parameter in advance. As an alternative, Jeyaratnam (1985) presented a theorem whose application yields a confidence region with predetermined confidence coefficient, with the benefit that no allocation has to be made in advance. Additionally, the resulting confidence region has minimum volume among all confidence regions that are based on the same pivot statistic. So far, there are only a few applications of this theorem. In this thesis, we use the theorem of Jeyaratnam to determine multidimensional confidence regions for the parameters of the Pareto distribution, the two-parameter exponential distribution, and the normal distribution. For one sample of Pareto data, joint confidence regions for the parameters are determined in case of a complete, a type-II right censored, and a doubly type-II censored sample. These confidence regions are compared in terms of shape and volume to those found in the literature by means of some simulations. Furthermore, we consider the case where two independent samples of Pareto data are available. Joint confidence regions for several different situations like for example partially known parameters or common parameters are presented. Subsequently, some of the results are applied to a real data set. Concerning the two-parameter exponential distribution we focus in the one sample case on a type-II right censored sample and a doubly type-II censored sample, since a minimum volume confidence region based on a complete sample can already be found in the literature. For both types of censoring minimum volume confidence regions are obtained. For the confidence region based on the type-II right censored sample a comparison in terms of shape and volume is made to a confidence region found in the literature and to one obtained by a standard procedure. These three confidence regions are then taken to address briefly coverage probabilities of false parameters, which may serve as another quality measure of confidence regions. Then we turn to the case of two independent samples of exponential data and we determine minimum volume confidence regions of two, three, and four dimensions followed by an application to a real data set. For the normal distribution we focus on the case of two independent samples. Several different situations are considered, and in all of them minimum volume confidence regions are obtained. Moreover, we present for a known covariance matrix a minimum volume confidence region for the mean vector of a multivariate normal distribution. Additionally, for all three considered distributions we present several plots of the obtained confidence regions and we provide closed formulas for their volumes in some cases., Published by Aachen

This PhD thesis deals with a new concept for a statistic course in engineering sciences resulting in a blended-learning structure. The manner of presenting the content intends to promote a strong practical reference basing on DIN Standards and VDI Guidelines. Out of the large number of partly non-networked Standards and Guidelines, in which furthermore other terms are used as in other aereas of statistic contents, resulted a systematic and practice-oriented elaboration of the contents belonging to inferential statistical analysis devided in the areas of point estimation, interval estimation and hypothesis testing. Real life examples form a very important component that should achieve situational learning with a constructivist approach. The aspired blended-learning concept is didacitcallyconsolidated. This fact underlining, several instruments are presented and embedded in the statistical context, such as the Myers Briggs type indicator, Kolb’s learning cycle or the teaching and learning methods by Felder/Silverman.In addition to the modified way of presentation of the contents, a change in the methodology is going to happen. Therefore the existing face-to-face-teaching course proceeds to a blended-learning course. Blended-learning describes a combination of face-to-face and online-/self-learning phases. It presents a chance to help students with different prior knowledge as well as to support individual learning preferences. Studying self-controlled through new designed online-courses by the teaching and learning environment EMILeAstat 2.0 offers in many ways greater flexibilty and above all the possibility of interaction, such as through integrated multiple-choice questions and especially developed applets. Within this thesis both technical as well as content details of the teaching and learningenvironment EMILeA-stat 2.0 are explicitly illustrated. A special focus lies on the developement of several applets by GeoGebra as an alternative to Java applets for the inferential statistical analysis. These should lead to an interactive and competence-oriented learning of the statistical contents. Additionally these applets should motivate an activeand intensive work. Finally this thesis includes two online learning units as components of a future blended-learning course.