# Contributions to statistical inference based on sequential order statistics from exponential and Weibull distributions

• Beiträge zur Statistischen Inferenz basierend auf Sequenziellen Ordnungsstatistiken aus Exponential- und Weibullverteilungen

Johnen, Marcus; Kamps, Udo (Thesis advisor); Kateri, Maria (Thesis advisor)

Aachen (2020)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2020

Abstract

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.