Generelized Algebraic Bounds on Order Statistics Functions, with Application to Reinsurance and Catastrophic Risk

Generalized algebraic bounds on linear combinations of order statistics are derived. Examples include average upper order statistics, Gini’s mean difference and stop-loss statistics. For the latter, different bounds are obtained and compared. In particular, it is shown that for the extreme values of the standardized order statistics derived by Hawkins(l971), an algebraic bound on the standardized stop-loss statistic evaluated at an arbitrary order statistic improves always on the corresponding generalized algebraic bound obtained from the sample version of the inequality of Bowers(1969). Furthermore, a new elementary constructive proof of Hawkins’ bounds is added. Actuarial applications to reinsurance and catastrophe risk are discussed.

KEYWORDS: algebraic boundedness, order statistics, stop-loss statistics, Hawkins’ inequality, Bowers’ inequality, Gini’s mean difference, reinsurance, catastrophe risk, extreme value theory, generalized Pareto