Institut für Angewandte Mathematik und Statistik

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Test for the model selection from two competing distribution classes
(2016) Chen, Hong; Jensen, Uwe
One 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.

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