Maik Döring

Estimation of the lifetime distribution of mechatronic systems in the presence of a covariate: A comparison among parametric, semiparametric and nonparametric models

Abstract In practice manufacturers may have lots of failure data of similar products using the same technology basis under different operating conditions. Thus, one can try to derive predictions for the distribution of the lifetime of newly developed components or new application environments through the existing data using regression models based on covariates. Three categories of such regression models are considered: a parametric, a semiparametric and a nonparametric approach. First, we assume that the lifetime is Weibull distributed, where its parameters are modelled as linear functions of the covariate. Second, the Cox proportional hazards model, well-known in Survival Analysis, is applied. Finally, a kernel estimator is used to interpolate between empirical distribution functions. In particular the last case is new in the context of reliability analysis. We propose a goodness of fit measure (GoF), which can be applied to all three types of regression models. Using this GoF measure we discuss a new model selection procedure. To illustrate this method of reliability prediction, the three classes of regression models are applied to real test data of motor experiments. Further the performance of the approaches is investigated by Monte Carlo simulations.

Exact Rate of Convergence of Kernel-Based Classification Rule

A binary classification problem is considered, where the posteriori probability is estimated by the nonparametric kernel regression estimate with naive kernel. The excess error probability of the corresponding plug-in decision classification rule according to the error probability of the Bayes decision is studied such that the excess error probability is decomposed into approximation and estimation error. A general formula is derived for the approximation error. Under a weak margin condition and various smoothness conditions, tight upper bounds are presented on the approximation error. By a Berry-Esseen type central limit theorem a general expression for the estimation error is shown.

Change Point Estimation in Regression Models with Fixed Design

In this paper, we consider a simple regression model with change points in the regression function which can be one of two types: A so called smooth bent-line change point or a discontinuity point of a regression function. In both cases we investigate the consistency of the M-estimates of the change points. It turns out that the rates of convergence are n 1 ∕ 2 or n, respectively, where n denotes the sample size in a fixed design. In addition, the asymptotic distributions of the change point estimators are investigated.

Model Selection Using Cramér–von Mises Distance

In this paper we consider a model selection problem for the distribution function of lifetimes in the presence of covariates. We propose a new model selection method by defining the closeness between two distribution functions by the Cramer–von Mises distance. This distance is used mostly in the literature to conduct goodness of fit tests. Given a set of data and two competing classes of parametric distribution functions, we define a test statistic, to decide which class approximates the underlying distribution better. With increasing sample size the asymptotic normality property of our test statistic is shown under suitable conditions. As an example, we apply our method to a real data set of lifetimes of DC-motors, which depend on the covariate load.

Rate of Convergence of a Change Point Estimator in a Misspecified Regression Model

A parametric estimation problem is considered in a misspecified regression model, where the regression function has a smooth change point. The focus lies on regression functions, which are continuous at the change point. Here, it is not assumed that the true regression function belongs to the model class. However, there exists a pseudo change point, such that the related regression function gives a reasonable approximation. With increasing sample size the asymptotic behavior is investigated of the least squares estimates of the change point. The consistency of the change point estimator for the pseudo estimator is shown. It turns out that the rate of convergence depends on the order of smoothness of the regression function at the change point.

Mathematische Modelle zur quantitativen Analyse der Zuverlässigkeit

In diesem Kapitel werden zum einen die Regressionsmodelle der Lebensdaueranalyse vorgestellt. Zum anderen werden mathematische Modelle zur Beschreibung komplexer Systeme betrachtet, welche aus Komponenten zusammengesetzt sind. Mogliche Abhangigkeiten zwischen den Lebensdauern der Komponenten werden durch so genannte Copula-Modelle beschrieben, die es unter anderem ermoglichen, den Einfluss des Grades der Abhangigkeit auf die Zuverlassigkeit des Gesamtsystems zu untersuchen.

Grundlagen für eine Zuverlässigkeitsbewertung mechatronischer Systeme

In diesem Kapitel werden die Grundlagen fur eine Zuverlassigkeitsbewertung mechatronischer Systeme thematisiert. Diese reichen von den verschiedenen Zuverlassigkeitsanalysen, Definitionen von Fehlern und Ursachen, den Vorgehensweisen zur Bestimmung der Zuverlassigkeit in den jeweiligen Domanen bis hin zu Forderungen fur eine zu erstellende Methodik zur Zuverlassigkeitsbewertung mechatronischer Systeme in fruhen Entwicklungsphasen.