# Divergence measures for generalized order statistics

• Divergenzmaße für verallgemeinerte Ordnungsstatistiken

Vuong, Quan Nhon; Kamps, Udo (Thesis advisor)

Aachen : Publikationsserver der RWTH Aachen University (2012, 2013)
Dissertation / PhD Thesis

Aachen, Techn. Hochsch., Diss., 2012

Abstract

The dissertation deals with the quantification of the differences between various models for ordered random variables. For this, generalized order statistics are considered, with which many different of such models can be described by appropriate choice of parameters (e.g., ordinary order statistics, sequential order statistics, record values, Pfeifer record values, and progressively Type II censored order statistics). For generalized order statistics, formulas for various divergence and distance measures (such as Kullback-Leibler divergence, Jeffreys distance, and Hellinger distance) are derived. Using the latter, the dissimilarity of different distributions for specific model parameter choices can be quantified. For the derivation of the formulas the exponential family structure of the distributions of generalized order statistics is exploited. Using the explicit representations of the divergence and distance measures, which have some remarkable properties in common, relations between different models are identified and named. For example, to calculate the divergences and distances only the componentwise model parameter ratios corresponding to the respective distributions are required, which, for example, leads to some structural similarities between models of sequential order statistics and Pfeifer record values. Moreover, the underlying distribution of the ordered random variables is irrelevant for the divergences and distances. Further results are obtained by deriving and analyzing models of order statistics which are closest (with respect to different divergence and distance measures) to a given model of sequential order statistics. Thus, an alternative modeling with ordinary order statistics can be considered if it is advantageous over a model with sequential order statistics for one’s purposes. In addition, the different models are identified with their parameter vectors, and interpreted and presented as points in Euclidean space. In this context, spheres and balls with respect to different divergence and distance measures are investigated and illustrated. As examples of applications that are directly enabled by the explicit representations of the divergence and distance measures for models of generalized order statistics, multi-dimensional confidence regions, homogeneity test for t> 2 populations, and a new point estimation method are presented. The implicit confidence regions are empirically compared with respect to their volumes using simulations. The description of homogeneity tests refers to asymptotic tests for exponential families which are based on a divergence measure and can be found in the literature. Such a test is compared for an example of sequential order statistics with the corresponding exact test. The new approach equal distance estimation is based on the maximum likelihood estimator and a pre-estimate using some prior information about the parameter to be estimated. It requires equal distance of the equal distance estimator to both other estimates. First studies demonstrated for the case with one parameter that there are situations in which the equal distance estimator is preferable to both, the maximum likelihood estimator and the pre-estimate. A performed simulation study indicates an even greater potential in the multi-parameter case.