“Bivariate Extreme Statistics, I”, 1960 (; backlinks; similar):
The largest and the smallest value in a sample, and other statistics related to them are generally named extreme statistics. Their sampling distributions, especially the limit distributions, have been studied by many authors, and principal results are summarized in the recent Gumbel book.
The author extends here the notion of extreme statistics into bivariate distributions and considers the joint distributions of maxima of components in sample vectors. This Part I treats asymptotic properties of the joint distributions.
In the univariate case the limit distributions of the sample maximum were limited to only three types. In the bivariate case, however, types of the limit joint distribution are various: Theorem 5 in Chapter 2 shows that infinitely many types of limit distributions may exist. For a wide class of distributions, two maxima are asymptotically independent or degenerate on a curve. Theorems 2 and 4 give the attraction domains for such limits. In bivariate normal case, two maxima are asymptotically independent unless the correlation coefficient is equal to one.
Throughout these arguments we remark only the dependence between marginal distributions, whose behaviors are well established. For this purpose a fundamental notion of ‘dependence function’ is introduced and discussed in §1.
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