Consistency and normality of estimates of hydrological extremes derived from a model for fragmented data

SummaryThis paper deals with a non-standard application of frequency analysis of extremes of hydrologic variables in which some years of record are incomplete. If F(y) is the cumulative distribution function (cdf) of the extreme variable in a complete year (or ‘block’), the cdf of the variable in an incomplete year is taken as F(y)p, where 0 < p < 1 is the proportion of observations existing for that year. In this model, which can also take into account whether the missing observations occurred in the wet or dry season, the extreme values in different years no longer have the same parent distribution, so that where parameters of F(y) are estimated by maximum likelihood (ML), the usual large-sample characteristics of ML estimators (consistency, asymptotic Normality) may be modified. The paper examines the consistency, bias and approach to Normality of ML estimates of Gumbel parameters for position and scale, and of the Gumbel extreme event y100 with 100-year return period. In the non-standard model F(y)p, consistency of ML estimates of Gumbel position and scale parameters is considerably modified, but the consistency of the estimated y100 much less so. Estimates of y100 are negatively biased, but the bias is similar to that found in the standard (no missing data) case. The results are relevant where hydrologic records are short and incomplete, such that all existing data must be fully utilized.