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Combinatorial extreme value distributions

Published online by Cambridge University Press:  26 February 2010

F. N. David
Department of Statistics, University College, London, W.C.1.
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Extreme value problems and particularly those arising in the combinatorial field have a peculiar interest and challenge in that an exact solution is rarely possible. We discuss here four combinatorial extreme value problems each concerned with the distribution of the largest (or smallest) of a set of mutually dependent variables. These problems, widely different in character possess three points in common. First the probability distribution functions of the variables considered cannot be obtained explicitly, nor secondly can the moments of the variable, and thirdly the probability distribution functions are difficult to evaluate even for moderate sized samples. We shall treat the situation where the variables, or functions of them, have probability distribution functions which tend to an exponential limit, analogous to the well-known limit for extreme values in the case of independent events. The p.d.f.'s for the upper tails of the distributions which we consider are found to be very closely contained within a pair of inequalities (Bonferroni Inequalities), especially in the regions of statistical significance.

Research Article
Copyright © University College London 1959

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