Multidimensional confidence regions for pareto, exponential, and normal distributions

• Mehrdimensionale Konfidenzbereiche für Pareto-, Exponential- und Normalverteilungen

Lennartz, Jens; Kamps, Udo (Thesis advisor); Kateri, Maria (Thesis advisor)

Aachen (2017)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2017

Abstract

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.