Unbiased plotting positions for the general extreme value distribution

Abstract

Probability plots maintain their popularity for flood frequency analysis, and are used to fit distributions, to identify outliers and to assess goodness-of-fit. If the objective of a plot is to determine quantiles or distribution parameters then one must employ unbiased plotting positions, defined as the mean of the rth order statistic in samples from the reduced variate population. These plotting positions differ from one distribution to another. The general extreme value (GEV) distribution is commonly used for flood frequency analysis, and contains the Gumbel distribution as a special case. Whilst plotting position formulae for this special case are well established, results are not available for the general case. By careful sequencing of the individual terms it is possible to integrate numerically the basic expression for expected order statistics to compute plotting positions for all practical sample sizes. However this integration consumes considerable computer resources, and for sample sizes of 35 and less exact unbiased plotting positions can be more readily evaluated using expressions derived using probability weighted moment theory. Numerical problems force an alternative approach for larger samples, and an asymptotic approximation has been developed. This has been found to be very accurate even for small samples, although an exact formula should be used for the largest flood in a sample. An even simpler level of calculation may be appropriate in some circumstances, and plotting position formulae have been developed based on expressions for the non-exceedance probability of the form F r = (r − α) (n − β) . The parameters α and β are allowed to vary with sample size and the shape parameter of the general extreme value distribution.