Extreme value theory and its potential role in chemometrics: An introduction

Abstract

In statistics, distributions are studied in their ‘normal’ ranges. ‘Normal’ stands for close to the mean and a few standard deviations away from the mean. Other values are considered as outliers. Outliers are not always bad, provided that they can be treated in an appropriate way, but some values that are measured are simply not outliers, just extremely large or small compared with the bulk of the samples and usually quite rare. A little known and quite different field in statistics is the one where the tails of distributions are considered. This is important in insurance studies (actuarial statistics), in engineering, mostly mechanical and construction engineering, and in the study of natural processes. This field is called ‘extreme value theory’. The main point of extreme value theory is that the tail of a distribution can behave quite differently from the gross of the population members. Systematic ways of describing tails and specific estimation procedures for tail distribution parameters have been developed during this century and are finding use in engineering and research applications. A definition and a historical overview of extreme value theory are given and some specific useful distributions are described. The estimation procedure and interpretation of the results are given for some classical examples from engineering. Important aspects such as visualization in plots and parameter estimation are given. An industrial example is introduced. It involves the measurement of sizes and shapes of particles of a chemical. The paper is meant to introduce the concepts, definitions and basic ideas of extreme value theory into chemometrical circles where there is a great potential for their use. The explanations are kept on a simple intuitive level. Extreme value theory can be used in chemometrics and related fields for solving problems in environmental measurement, particle size distributions and industrial process control. © 1996 by John Wiley & Sons, Ltd.