Estimating trends in data from the Weibull and a generalized extreme value distribution

Abstract

Where changes in hydrologic regime occur, whether as a result of change in land use or climate, statistical procedures are needed to test for the existence of trend in hydrological data, particularly those expected to follow extreme value distributions such as annual peak discharges, annual minimum flows, and annual maximum rainfall intensities. Furthermore, where trend is detected, its magnitude must also be estimated. A later paper [Clarke, 2002] will consider the estimation of trends in Gumbel data; the present paper gives results on tests for the significance of trends in annual and minimum discharges, where these can be assumed to follow a Weibull distribution. The statistical procedures, already fully established in the statistical analysis of survival data, convert the problem into one in which a generalized linear model is fitted to a power-transformed variable having Poisson distribution and calculates the trend coefficients (as well as the parameter in the power transform) by maximum likelihood. The methods are used to test for trend in annual minimum flows over a 19-year period in the River Paraguay at Cáceres, Brazil, and in monthly flows at the same site. Extension of the procedure to testing for trend in data following a generalized extreme value distribution is also discussed. Although a test for time trend in Weibull-distributed hydrologic data is the motivation for this paper, the same approach can be applied in the analysis of data sequences that can be regarded as stationary in time, for which the objective is to explore relationships between a Weibull variate and other variables (covariates) that explain its behavior.