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An introduction is provided to the classical theory of asymptotic distributions of extreme values and to its application to estimation of tail probabilities. It is shown how the theory is sometimes misinterpreted to justify heuristic estimation procedures that introduce systematic error. A new generalized asymptotically convergent sequence of extreme-value distributions, differing from the classical sequence in higher order terms, which eventually disappear, is derived for the "exponential'' class of distributions. It is shown that in some cases of practical interest, the generalized sequence converges much faster to the asymptote than does the widely used classical sequence, and that tail-probability estimates of a given accuracy can be formulated from a much smaller number of data.