Robustness of tail index estimation.

One of the important issues in the study of long-tailed, or outlier-prone, probability distributions is the estimation of the tail index $\alpha,$ which measures the heaviness of the tail of a distribution. A simple tail index estimator proposed by Hill (1975) has been widely implemented in many professional fields where risk assessment is important. The implementation of the Hill estimator, however, requires a choice of the number of observations in the tails, r, from a sample of size n, where $r\le n.$ Several procedures proposed in the literature for choosing r rely upon one or more strong assumptions about the underlying distribution of the data or overlook the effect of the sample size. This dissertation is concerned with robust procedures of choosing an optimal r. Two estimation procedures are developed: one based on the idea of spacing statistics, $H\sp{(r)}$ and $K\sp{(r)},$ wherein a knowledge of the underlying global distribution is not required; and the other that combines the spacing statistic $H\sp{(r)}$ and a nonparametric density model. The proposed decision rules for choosing r under various loss functions are found to be simple functions of sample size. These rules are then shown to be effective across a wide range of data, including city size, insurance claims, and currency exchange rate returns.Ph.D.Pure SciencesStatisticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/130261/2/9722001.pd