Udo Kamps

Udo Kamps
RWTH Aachen University · Institute of Statistics

Professor

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199
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Introduction
Udo Kamps currently works at the Institute of Statistics, RWTH Aachen University. Udo does research in Statistics and Stochastic Modelling.

Publications

Publications (199)
Chapter
Parameter estimation is discussed when the underlying parametric model forms an exponential family. For maximum likelihood estimates, the topics existence, uniqueness, representation, and properties are addressed, and constrained maximum likelihood estimation is examined as well. Efficient estimation in exponential families and asymptotic propertie...
Chapter
In a one-parameter exponential family, uniformly most powerful tests are stated for particular, but important hypotheses. Moreover, uniformly most powerful unbiased tests are shown in other cases and in multiparameter exponential families. Statistical tests for several parameters simultaneously are considered with a focus on likelihood-ratio tests,...
Chapter
With a focus on statistical inference, distributional and statistical properties of multivariate and multiparameter exponential families are studied, such as generating functions, marginal and conditional distributions, and product measures. Sufficiency and completeness of statistics in exponential families is examined, and representations of the s...
Chapter
A formal definition of an exponential family is the starting point of a widely self-contained, mathematically rigorous and concise course on exponential families in a systematic structure. The definition is followed by a variety of detailed examples of one- and multiparameter, uni- and multivariate probability distributions. Basic notions and repre...
Chapter
The usefulness of results for exponential families is demonstrated by means of three multivariate examples of different kind. As a discrete probability distribution, the family of negative multinomial distributions is considered, and results in parameter estimation and statistical testing are shown. The Dirichlet distribution serves as an example o...
Chapter
Parametric families of probability distributions and their properties are extensively studied in the literature on statistical modeling and inference. Exponential families of distributions comprise density functions of a particular form, which enables general assertions and leads to nice features. With a focus on statistical inference, a motivation...
Article
Point prediction of future record values from a sequence of independent and identically distributed Pareto and Lomax random variables is addressed. The focus is on likelihood-based prediction techniques; in particular, the maximum likelihood as well as the maximum observed likelihood prediction principles are invoked to derive predictors. Pitman cl...
Article
Full-text available
Within exponential families, which may consist of multi-parameter and multivariate distributions, a variety of divergence measures, such as the Kullback–Leibler divergence, the Cressie–Read divergence, the Rényi divergence, and the Hellinger metric, can be explicitly expressed in terms of the respective cumulant function and mean value function. Mo...
Article
Full-text available
Estimation of model parameters of sequential order statistics under linear and nonlinear link function assumptions is considered. Utilizing the arising curved exponential family structure, conditions for existence and uniqueness as well as the validity of asymptotic properties of maximum likelihood estimators are stated. Minimal sufficiency and com...
Article
Full-text available
Based on a doubly type-II censored sample, a joint confidence region with minimum area is provided for the location and scale parameter of the exponential distribution. In the type-II right censored case, explicit formulas for the associated (expected) minimum area and the coverage probabilities of false parameters are derived. For both quality mea...
Book
This book provides a concise introduction to exponential families. Parametric families of probability distributions and their properties are extensively studied in the literature on statistical modeling and inference. Exponential families of distributions comprise density functions of a particular form, which enables general assertions and leads to...
Article
Capacity degradation of lithium-ion batteries under long-term cyclic aging is modeled via a flexible sigmoidal-type regression setup , where the regression parameters can be interpreted. Different approaches known from the literature are discussed and compared with the new proposal. Statistical procedures, such as parameter estimation, confidence a...
Article
Full-text available
A spacings-based prediction method for future upper record values is proposed as an alternative to maximum likelihood prediction. For an underlying family of distributions with continuous cumulative distribution functions, the general form of the predictor as a function of the estimator of the distributional parameters is established. A connection...
Article
Interval prediction of future generalized order statistics from exponential distributions based on a multiply Type-II censored sample is considered. On invoking the principle of reduction by equivariance, prediction regions of minimal size are derived and shown to be of interval form. An alternative derivation based on the use of a prediction suffi...
Article
Link functions of distributional and model parameters, such as linear, log-linear, or inverse linear functions, are commonly utilized in accelerated life testing to include expert knowledge and to reduce the number of unknown parameters. Although, there might be some evidence in choosing a specific link function in the analysis of an experiment, it...
Article
Full-text available
Pfeifer record values are based on a double sequence of independent, non-identically distributed random variables. For Pfeifer records with a common baseline distribution function and depending on a sequence of positive parameters, almost sure convergence to a non-degenerate random variable is examined, and sufficient conditions for almost sure rel...
Article
Independent samples from different exponential distributions are available in many statistical applications, for example, in queuing or reliability models, where the random variables describe waiting times and lifetimes, respectively. When two samples are observed, inference is then carried out for two location and two scale parameters. In multivar...
Article
Full-text available
A multi-sample set-up of sequential order statistics from Weibull distribution functions with known scale parameters and a common unknown shape parameter is considered. The respective likelihood equation may have multiple roots even in the single-sample case, which is demonstrated by a simple example and illustrated with a simulation study. Uniquen...
Article
In lifetime analysis of lithium-ion batteries, the failure of a battery is commonly defined as the time point at which the battery holds only a certain percentage of its initial storage capacity. Usually, lifetime experiments are carried out to record degradation up to a certain level, and thus do not offer an appropriate data base for fitting life...
Chapter
In stochastischen Modellen werden interessierende Merkmale (vgl. auch hierzu den Begriff „Merkmal“ in Abschnitt A 1.2) in der Regel mit Zufallsvariablen beschrieben. Dies sind Abbildungen von einem zugrundeliegenden Wahrscheinlichkeitsraum in einen neuen Wahrscheinlichkeitsraum, der einerseits eine einfachere Struktur hat und andererseits eine „gut...
Chapter
Modelle für reale, zufallsabhängige Vorgänge werden in vielen Bereichen von Wissenschaft, Technik und Wirtschaft eingesetzt. Diese dienen der (vereinfachten) Beschreibung der Wirklichkeit und dem Zweck, Aussagen im Modell zu gewinnen. Diese Ergebnisse können, falls das Modell „gut genug“ ist, durch „Rückübersetzung“ in die Realität Entscheidungshil...
Chapter
Eine Darstellung der Beschreibenden Statistik erfordert Motivationen, Beispiele, Erläuterungen, Interpretationen, Nachweise und graphische Umsetzungen bzw. Illustrationen, für die in diesem Manuskript nicht immer genügend Raum ist. Eine ausführliche Darstellung findet sich im Lehrbuch zur Beschreibenden Statistik Burkschat et al. (2012).
Chapter
In den Multiple-Choice-Fragen wird nach wahren und falschen Aussagen gefragt. In den Hinweisen zu den Antwortalternativen wird aber jeweils der Wahrheitsgehalt der entsprechenden (einzelnen) Aussage wie folgt gekennzeichnet:
Chapter
Bei der Formulierung von (1−α)-Konfidenzintervallen und α-Niveau-Tests werden Quantile zur Festlegung von Intervallgrenzen bzw. von Entscheidungsregeln der Hypothesentests verwendet (s. Kapitel D 4-D 7).
Chapter
Mit den Mitteln der Deskriptiven Statistik werden für eine Gruppe von Objekten Eigenschaften eines Merkmals dieser Objekte anhand von Beobachtungswerten beschrieben. Alle Aussagen beziehen sich ausschließlich auf die zugrundeliegenden Objekte und die für diese beobachteten Werte.
Article
Full-text available
Point prediction of future upper record values is considered. For an underlying absolutely continuous distribution with strictly increasing cumulative distribution function, the general form of the predictor obtained by maximizing the observed predictive likelihood function is established. The results are illustrated for the exponential, extreme-va...
Book
Dieses Buch führt anwendungsorientiert in die Beschreibende und Schließende Statistik, in die Wahrscheinlichkeitsrechnung und in die Stochastische Modellierung ein und wendet sich insbesondere an Studierende der Informatik, des Ingenieur- und Wirtschaftsingenieurwesens sowie der Wirtschaftswissenschaften. Es ist ein idealer Begleiter zu jeder einse...
Article
In a multi-sample model of multivariate normal distributions with covariance matrices being known or known except for unknown multipliers, simultaneous confidence regions for the mean vectors are provided with minimum volume properties. The univariate case with unknown variances is included.
Article
Whereas discrimination methods and their error probabilities were broadly investigated for common data distributions such as the multivariate normal or t-distributions, this paper considers the case when the recorded data are assumed to be observations from sequential order statistics. Random vectors of sequential order statistics describe, e.g., s...
Preprint
Capacity degradation of lithium-ion batteries under long-term cyclic aging is modelled via a flexible sigmoidal-type regression set-up, where the regression parameters can be interpreted. Different approaches known from the literature are discussed and compared with the new proposal. Statistical procedures, such as parameter estimation, confidence...
Article
Based on independent samples of progressively type-II censored order statistics, a two-dimensional confidence region for the parameters of an underlying distribution out of a location–scale family is provided, which has smallest area within a wide class of confidence regions.
Article
Characterizing relations via Rényi entropy of m-generalized order statistics are considered along with examples and related stochastic orderings. Previous results for common order statistics are included.
Article
Full-text available
In reliability, sequential order statistics serve as a model for the component lifetimes of k-out-of-n systems, which are operating as long as k out of n components are operating. In contrast to modelling with order statistics, load-sharing effects and other impacts of failures on the performance of the remaining components may be taken into consid...
Article
[Abstract] Based on a single and on two independent samples, joint confidence regions for parameters of Pareto distributions are proposed with minimum volume properties and without assigning the confidence level to dimensions. In the one-sample case, comparisons are made to former simultaneous confidence sets for Pareto parameters by means of simul...
Article
In a set-up of independent samples of sequential order statistics from Weibull distributions, which may describe a load-sharing lifetime experiment, statistical inference is considered for scale parameters and a common shape parameter. A representation of the corresponding Fisher information matrix is stated, and the joint maximum likelihood estima...
Article
In a multi-sample general step–stress model with pre-specified numbers of observations under all stress levels, confidence regions for associated parameters are provided with minimum volume, minimum coverage probabilities of false parameters, or based on divergence measures.
Chapter
Die in diesem Kapitel enthaltenen Inhalte stehen Ihnen als Download unter www.toolbox.cemath.rwth-aachen.de kostenfrei zur Verfügung, sind aber nicht Bestandteil des Buchs.
Article
Step-stress models form an essential part of accelerated life testing procedures. Under a step-stress model, the test units are exposed to stress levels that increase at intermediate time points of the experiment. The goal is to develop statistical inference for, e.g., the mean lifetime under each stress level, targeting to the extrapolation under...
Article
As a submodel of generalized order statistics with two unknown model parameters, m-generalized order statistics may serve as a simple model for ordered quantities in a given application. It is shown that the joint distribution of m-generalized order statistics has a representation as a regular exponential family in the model parameters, as it is th...
Chapter
Stochastische Modelle werden in vielen Bereichen von Wissenschaft, Technik und Wirtschaft zur Beschreibung realer, zufallsabhängiger Vorgänge eingesetzt. Diese dienen der (vereinfachten) Beschreibung der Wirklichkeit und dem Zweck, Aussagen im Modell zu gewinnen. Ist eine hinreichende Übereinstimmung des Modells mit der Wirklichkeit gegeben, so kön...
Chapter
Statistische Grundkenntnisse sind in einer von Daten geprägten Zeit von besonderer Bedeutung. In vielen Bereichen von Wirtschaft, Wissenschaft, Verwaltung, Gesellschaft und Politik werden Informationen aus Daten gewonnen, die empirischen Studien, Datenbasen, Erhebungen oder Experimenten entstammen. Informationen werden in quantitativer Weise verbre...
Chapter
Bei der Formulierung von (1−α)-Konfidenzintervallen und α-Niveau-Tests werden Quantile zur Festlegung von Intervallgrenzen bzw. von Entscheidungsregeln der Hypothesentests verwendet (s. Kapitel D 4-D 7). Für die im Rahmen der Normalverteilungsmodelle auftretenden Verteilungen, d.h. für die Standardnormal, t(n)-, χ2(n)- und F(n,m)-Verteilung, werden...
Chapter
Mit den Mitteln der Deskriptiven Statistik werden für eine Gruppe von Objekten Eigenschaften eines Merkmals dieser Objekte mittels Kenngrößen und graphischen Verfahren analysiert, aufbereitet und präsentiert. In der Schließenden Statistik (oder Induktiven Statistik) werden Verfahren und Methoden bereitgestellt, die es ermöglichen, auf der Basis sta...
Chapter
Stochastische Modelle werden in vielen Bereichen von Wissenschaft, Technik und Wirtschaft zur Beschreibung realer, zufallsabhängiger Vorgänge eingesetzt. Diese dienen der (vereinfachten) Beschreibung der Wirklichkeit und dem Zweck, Aussagen im Modell zu gewinnen. Ist eine hinreichende Übereinstimmung des Modells mit der Wirklichkeit gegeben, so kön...
Book
Dieses Buch führt anwendungsorientiert in die Beschreibende und Schließende Statistik, in die Wahrscheinlichkeitsrechnung und in Stochastische Modellierung ein und wendet sich insbesondere an Studierende der Informatik, des Ingenieur- und Wirtschaftsingenieurwesens sowie der Wirtschaftswissenschaften. Es ist als begleitender Text zu einer einsemest...
Article
Die Mathe-Toolbox bietet ein zielgerichtetes Mathematik-Training und Unterstützung beim Übergang Schule-Hochschule sowohl für Studienanfängerinnen und -anfänger in MINT-Studiengängen (Ingenieurwesen, Informatik, Naturwissenschaften, Mathematik und weitere mathematiknahe Studiengänge) als auch für Schülerinnen und Schüler in der Studienorientierungs...
Article
An extended model of order statistics based on possibly different distributions is introduced and analyzed. In the interpretation of successive failure times in a 𝑘-out-of-𝑛 system, say, until each failure, the time periods under previous (increasing) loads exerted on the remaining components are recorded. Then the lifetime distribution of the syst...
Chapter
Order statistics and record values appear in many statistical applications and are widely used in statistical modeling and inference. Both models describe random variables arranged in ascending order of magnitude. Generalized order statistics provide a unified approach to a variety of models of ordered random variables with different interpretation...
Article
A classification procedure for a two-class problem is introduced and analyzed, where the classes of probability density functions within a regular exponential family are represented by left-sided Kullback–Leibler balls of natural parameter vectors. If the class membership is known for a finite number of densities, only, classes are defined by const...
Article
Based on stochastically independent samples with underlying density functions from the same multiparameter exponential family, a weighted version of Matusita’s affinity is applied as test statistic in a homogeneity test of identical densities as well as in a discrimination problem. Asymptotic distributions of the test statistics are stated, and the...
Article
In applications of generalized order statistics as, for instance, reliability analysis of engineering systems, prior knowledge about the order of the underlying model parameters is often available and may therefore be incorporated in inferential procedures. Taking this information into account, we establish the likelihood ratio test, Rao's score te...
Article
In the analysis of maximum sizes of large non-metallic inclusions in steels, the common extreme value analysis is only based on a sample of largest observations in control areas on polished planes, respectively. From a simulation study it is observed that this univariate set-up may lead to a high proportion of misspecifications of the true extreme...
Article
A multi-sample model for general step-stress experiments based on sequential order statistics is proposed and analyzed, where the numbers of observations under each stress level are pre-specified. Contrary to common step-stress approaches, an arbitrary absolutely continuous lifetime distribution can be chosen, and, due to the experimental design of...
Article
For successive failure times of components in a technical system, a flexible model based on sequential order statistics is proposed. Beyond the common assumption of proportionality, this model allows for structural adjustments of the hazard rates of the underlying lifetime distributions in situations, where failures have an impact on the entire sha...
Chapter
As a generalization of order statistics from independent and identically distributed random variables, sequential order statistics (SOSs) may be applied as a model for ordered data, when assuming changes of underlying distributions immediately after the occurrences of ordered observations. For example, in the case of a model for k-out-of- n-systems...
Article
In goodness-of-fit testing based on Type-II right censored samples, mainly either classical tests are applied to transformed samples, which can be considered as full samples under the null hypothesis, or modifications of the tests are applied to the original samples. Results of an extensive simulation study are presented and discussed in order to c...
Article
The exponential family structure of the joint distribution of generalized order statistics is utilized to establish multivariate tests on the model parameters. For simple and composite null hypotheses, the likelihood ratio test (LR test), Wald's test, and Rao's score test are derived and turn out to have simple representations. The asymptotic distr...
Chapter
Mit den Mitteln der Deskriptiven Statistik werden für eine Gruppe von Objekten Eigenschaften eines Merkmals dieser Objekte anhand In der Schließenden Statistik (oder Induktiven Statistik) werden Verfahren und Methoden bereit gestellt, die es ermöglichen, auf der Basis statistischer Modelle und Daten aus Stichproben zu allgemeinen Aussagen über eine...
Chapter
Statistische Grundkenntnisse sind in einer von Daten geprägten Zeit von besonderer Bedeutung. In vielen Bereichen von Wirtschaft, Wissenschaft, Verwaltung, Gesellschaft und Politik werden Informationen aus Daten gewonnen, die empirischen Studien, Datenbasen, Erhebungen oder Experimenten entstammen. Informationen werden in quantitativer Weise verbre...
Article
In step-stress modeling with the cumulative exposure model it is well known that, for underlying exponential distributions, explicit expressions can be obtained for maximum likelihood estimators of parameters as well as for their conditional density functions or conditional moment generation functions, given their existence. Applying a failure rate...
Article
Motivated from extreme value (EV) analysis for large non-metallic inclusions in engineering steels and a real data set, the benefit of choosing a multivariate EV approach is discussed. An extensive simulation study shows that the common univariate setup may lead to a high proportion of mis-specifications of the true EV distribution, as well as that...
Article
In a scheme of independent, possibly Type-II censored samples, joint estimation of scale parameters is considered under an order statistics prior leading to strictly ascendingly ordered Bayes estimators. The focus is on deriving handy closed-form expressions for the estimators.
Article
The situation of identical distributions of a single m-generalized order statistic and a random convex combination of two neighboring m-generalized order statistics with beta-distributed weights is studied, and related characterizations of generalized Pareto distributions are shown.
Article
The concept of generalized order statistics is a distribution theoretical set-up, which contains a variety of models for ordered data as particular cases, such as common order statistics, sequential order statistics, progressively type-II censored order statistics, record values, kth record values, and Pfeifer record values. In order to quantify th...
Article
As a particular structure in reliability, a sequential k-out-of-n system fails if more than of its n components fail, where the failure of some component may affect the residual lifetimes of the remaining components of the system. Sequential order statistics serve as a model for the (ordered) lifetimes of the components and the system, respectively...
Chapter
Beispiel Temperaturdaten In einer Stadt wurden in ausgewählten Monaten folgende monatlichen Durchschnittstemperaturen im Zeitraum 1996-2002 gemessen.
Chapter
Im ersten Anwendungsbeispiel 3▸Befragung der Mitarbeiterinnen eines Unternehmens werden einige grundsätzliche Gedanken zur Systematisierung und Auswertung einer statistischen Erhebung formuliert. Die resultierenden statistischen Grundbegriffe zur Beschreibung und Einordnung der interessierenden Größen einer Erhebung werden im ersten Kapitel erläute...
Chapter
Beispiel Produktion In einem Unternehmen sollen die Beschaffungsmengen und Beschaffungskosten der eingesetzten Rohstoffe sowie deren zeitliche Entwicklungen analysiert werden. Die beschafften Mengen der Rohstoffe A, B, C (in Tonnen) in den Jahren 1985 bis 2002 und die Rohstoffpreise (in €/Tonne) in den Jahren 1995 bis 2002 sind folgender Tabelle zu...
Chapter
Beispiel (Fortsetzung 3▸Beispiel Befragung der Mitarbeiterinnen) In der Personalabteilung des Unternehmens nahmen 15 Beschäftigte an der Fragebogenerhebung teil. Einen Ausschnitt des entstandenen Datensatzes für die Merkmale Pl Geschlecht (Kodierung: 0 männlich, 1 weiblich), P4 Dauer der Betriebszugehörigkeit (in Monaten), Bl Zufriedenheit mit dem...
Chapter
Beispiel Stadtfeste Für die Planungen der jährlichen Stadtfeste möchte eine Stadtverwaltung klären, wie stark der Zusammenhang zwischen der jeweiligen Besucherzahl und den Kosten für die Stadtreinigung ausgeprägt ist. Die Beziehung zwischen dem Merkmal X Besucherzahl und dem Merkmal X Reinigungskosten soll möglichst durch eine Funktion quantitativ...
Chapter
Beispiel Drogerieketten Die Drogeriekette A hat neben einem Stammhaus weitere 14 Filialen in anderen Städten. Die Geschäftsleitung möchte — zu ihrer Information und zum Vergleich mit Daten der Konkurrenz — im Rahmen einer statistischen Analyse der Umsätze (in Mio. €) des vergangenen Jahres in den Niederlassungen einige statistische Kenngrößen ermit...
Article
There are several well-known mappings which transform the first r common order statistics in a sample of size n from a standard uniform distribution to a full vector of dimension r of order statistics in a sample of size r from a uniform distribution. Continuing the results reported in a previous paper by the authors, it is shown that transformatio...
Article
An Opial-type inequality is applied to obtain relations for expectations of functions of m-generalized order statistics (m-gOSs), their distribution functions, as well as moment-generating functions. Respective inequalities for common order statistics and record values are contained as particular cases.
Article
For classical goodness-of-fit tests as well as for their modified versions in case of Type-II right censored data, results of a simulation study are presented, where the powers of the tests under consideration against beta alternatives are illustrated in informative, user-oriented plots that offer the possibility to directly choose the best test st...
Article
For general step-stress experiments with arbitrary baseline distributions, wherein the stress levels change immediately after having observed pre-specified numbers of observations under each stress level, a sequential order statistics model is proposed and associated inferential issues are discussed. Maximum likelihood estimators (MLEs) of the mean...
Article
Generalized order statistics, and thus sequential order statistics with conditional proportional hazard rates, are shown to form a regular exponential family in the model parameters. This structure is utilized to derive maximum likelihood estimators for these parameters or functions of them along with several properties of the estimators. The Fishe...
Article
Beispiel Reiseveranstalter In einem Marktsegment der Tourismusbranche konkurrierten im Jahr 2001 zehn Anbieter um die Gunst der reiselustigen Kunden. Die Umsätze (in Mio. €) des Jahres 2001 sind in der folgenden Tabelle zusammengestellt.
Article
Beispiel Geschwindigkeitsrnessung Eine Stadtverwaltung lässt das Verhalten der Verkehrsteilnehmerinnen gezielt und mit quantitativen Methoden untersuchen, um eine fundierte Grundlage zur Unterstützung von Entscheidungen zu erhalten. Dazu wurden beispielsweise am späten Abend Radarmessungen an einem Kontrollpunkt innerhalb der Stadt vorgenommen (zul...
Article
In a multi-sample experiment, we model the parame- ters of an equal load-sharing system by means of link functions in sequential order statistics models, and then discuss the estimation of these parameters based on a given link function. Different link functions are examined along with the corresponding maximum likelihood estimators, and their prop...
Chapter
Step-stress models are descriptions of special experiments in the field of accelerated life testing, where the test units are exposed to stress levels that change at intermediate time points during the experiment. The goal is to develop inference for the mean lifetime at each stress level. The time points of stress level change can either be fixed...
Article
Based on multiply Type-II censored samples of sequential order statistics, Bayesian estimators are derived for the parameters of one- and two-parameter exponential distributions. In the one-parameter set-up, the posterior density is obtained under the assumption that the prior distribution is given by an inverse Gamma distribution, and the Bayes es...
Article
In a Type-II right censored sample from the standard uniform distribution, several transformations of respective order statistics are examined, which transform the censored sample into a complete sample in a lower dimension. Such transformations have been considered by Lin et al. (2008), Michael and Schucany (1979) and O’Reilly and Stephens (1988)...
Article
In simple step-stress experiments under Type-II censoring with the cumulative exposure model and exponentially distributed lifetimes, maximum likelihood estimates (MLE) of the expected lifetimes may not exist due to the absence of failure times either before or after the stress change point. For this reason, when planning a step-stress experiment,...
Article
In the model of sequential order statistics, prior distributions are considered for the model parameters, which, for example, describe increasing load put on remaining components. Gamma priors are examined as well as priors out of a class of extended truncated Erlang distributions (ETED), which is introduced along with some properties. The choice o...

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