E. J. Gumbel

Bivariate Exponential Distributions

Abstract A bivariate distribution is not determined by the knowledge of the margins. Two bivariate distributions with exponential margins are analyzed and another is briefly mentioned. In the first distribution (2.1) the conditional expectation of one variable decreases to zero with increasing values of the other one. The coefficient of correlation is never positive and lies in the interval –.40≤ρ≤0, and the correlation ratio varies from –.48 to zero. In the second distribution (3.4) the conditional expectation of one variable increases or decreases with increasing values of the other variable depending on the sign of the correlation. The coefficient of correlation lies in the interval –.25≤ρ≤.25, and the correlation ratio is proportional to the coefficient.

Bivariate Logistic Distributions

Abstract The logistic distribution closely resembles the normal one. Both are symmetrical. Here two logistic bivariate distributions are studied. In both cases the curves of equal probability density are not ellipses, the regression curves are not linear and the conditional expectations are limited. The first distribution analyzed with the help of the bivariate moment generating function is asymmetrical and therefore departs considerably from the normal one. The coefficient of correlation is constant and equal to one half. The second bivariate logistic distribution is symmetrical. The regression curves are linear in probability scale and the coefficient of correlation varies in the interval ± .30396.

Some Analytical Properties of Bivariate Extremal Distributions

Abstract This article considers two examples of bivariate extremal distributions when the margins follow the first asymptotic distribution of the largest values. The variables are assumed to be reduced and each distribution contains only one parameter indicating the association between the extremes. The probability and the density functions of these distributions at the characteristic value, median, and mode have been analyzed. A procedure has been developed to estimate the single parameter of the distributions. A criterion for distinguishing between the two bivariate extremal distributions has been worked out. Finally, the various theories developed are applied to a numerical example.

Analysis of Empirical Bivariate Extremal Distributions

Abstract Two examples illustrate the use of bivariate extremal distributions of the first type. One is the distribution of oldest ages at death for the two sexes; the other consists of the floods of the same river recorded at two stations located upstream and downstream. The first is an illustration of independence; the second, of dependence. These illustrations were chosen so as to facilitate the decision on the hypothesis of independence. We limit our consideration to the first, double exponential, extremal distribution because it can be written in a form which is parameter-free. Distribution-free tests for the hypothesis of independence and a criterion for independence of bivariate extremal distributions are given. Bivariate density curves of independent largest values are shown.

Applications of the Circular Normal Distribution

Abstract * Work done in part as Consultant to Stanford University and in part under grant from the Higgins Foundation.

The return period of order statistics

It may be stressed that the results reached here are independent of the initial distribution and valid for any sample sizen and all order statistics. For example the expected return period for the smaller of two observations is 2, but infinite for the larger one and the variance of the return periods for both observations is infinite.

SOME APPLICATIONS OF EXTREME- VALUE METHODS

Classical applications of statistical methods, which frequently concern average values and other quantities following the symmetrical normal distribution, are inadequate when the quantity of interest is the largest or the smallest in a set of magnitudes. This is the situation in a number of fields, in many of which applications of methods for dealing with extremes have already been made. The literature on extreme value methods and applications is extensive. However, since it has largely developed in response to specific problems, it has until very recently been widely scattered in inumerous subject-matter publications and was therefore not readily accessible to the general research public. This limitation has now been removed with the publication by the National Bureau of Standards of Statistical Theory of Extreme Values and Some Practical Applications, a series of lectures [14]1 by one of the authors giving a comprehensive and unified account of the subject and including a comprehensive bibliography. A companion volume of tables was issued earlier [24]. The National Bureau of Standards has had a considerable interest in this subject, being concerned with the mathematical treatment of extreme values an-d the utilization of these and other data as a means of predicting future performance. One statistical program, originally sponsored by the National Advisory Committee for Aeronautics, was concerned with predicting the maximum gust velocities that aircraft structures would be likely to encounter [22]. The aim of this article is to presenit a brief survey of the nature of extreme-value methods and illustrate their use by several typical applications. The text is based primarily on material in Lectures 1 and 4 of [14], anid the first application is taken from Lecture 4 of the same source. These may be consulted for further details of the methods or applications referred to here. The field is constantly growing,2 and considerable additional work has already appeared since preparation of the lectures for publication was undertaken. Some of these are listed at the end of the article.

The calculated risk in flood control

A statistical function, the return period, will be linked to the calculated risk in order to obtain the value of the variable to be used in the design of a structure.