Extreme values of sampled and continuous stochastic data. Technical report No. 10

Statistical extreme value theory is obviously a potentially useful tool in the study of concentrations of materials associated with atmospheric pollution, when maximum levels are of primary interest. The classical theory requires the observed values to be statistically independent. Within the last decade there has been considerable interest in allowing a dependence structure, and an extensive extremal theory now exists for a wide class of statistically dependent values, measured as a discrete sequence, e.g., at specific sampling points. Of at least equal interest, however, are the statistical properties of the maxima of such quantities over continuous periods of time. For it is such a maximum--not necessarily occurring at a sampling point--which may lead to concern for effects on health, and so on. In the case of normal (Gaussian) data there has been considerable activity, and many explicit results are known. This report presents a general theory which applies to maxima in the continuous time context, for a wide class of basic distributions (normal and non-normal). It is found that the same basic results hold as for the sampled case--with certain important practical differences, which are described in detail