Estimating Changing Extremes Using Empirical Ranking Methods

It is often useful to make initial estimates of changing extremes without the use of a specific statistical model, though a statistical model is likely to be desirable as a second step. A proof is given of a formula used by A. F. Jenkinson in the 1970s that converts data that are ranked according to their magnitude into an estimate of the associated cumulative probability. This formula is compared to its exact equivalent, based on a beta distribution of the first kind. It is also compared to similar ranking formulas, which have been recommended, mostly in hydrology, based on similar ideas. Some results concerning the effect of serial correlation on Jenkinson's formula are reported. For initial estimates of return periods or percentiles of cumulative probability from time series of data, Jenkinson's method performs as well as many of the other methods. Empirical ranking methods are not so useful for estimating the rarest percentiles in climatology, those in the most extreme 100/N% tails of the distribution, say, where N is the data length. To estimate such extreme percentiles, distributional models are essential. However, for moderate extremes it is suggested that Jenkinson's or one of the similar methods are useful for an initial assessment of changing percentiles for a wide range of underlying data distributions.