Review of Statistics of Extreme Values with Applications to Air Quality Data Part I. Review

In the field of air pollution control, the rare event is often of more significance than the common event. This is evidenced by the content of air quality standards which define acceptable upper limits of air pollution concentrations and acceptable frequencies with which such concentrations can be exceeded. The principles of extreme value statistics provide important tools for analyzing air quality data in an appropriately significant context. The purpose of this paper is twofold. First the principles of extreme value statistics are reviewed. The presentation is decidedly utilitarian; no attempt is made to derive more than the simplest propositions. The underlying assumptions of the theory are carefully pointed out and discussed. Particular emphasis is placed on the second largest value and its relation to the largest value. Exact probability distributions, and approximations to them, are derived in terms of the total population probability function. The more familiar and useful asymptotic distributions and their uses are presented and discussed. Part II of the paper will present applications of the theory to air quality data.