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

Abstract

Summary This 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 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 y 100 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 y 100 much less so. Estimates of y 100 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.