Multivariate estimation of floods: the trivariate gumbel distribution

Abstract

Every year, floods take lives and damage properties in many parts of the World. A mean of reducing the damages caused by floods is with an adequate estimation of probable flows based on the probabilities of these events. For this purpose, a trivariate extreme value distribution have been derived from the logistic model for multivariate extreme value distributions. Due the complexity of the likelihood functions, the maximum likelihood estimators of the parameters of the trivariate distribution were obtained numerically by using a multivariable constrained non-linear optimization procedure which allows the cases of samples with different lengths of record. To investigate whether the estimates of the parameters based on trivariate procedures are better than those based on univariate procedures, the asymptotic variance—covariance matrices of the maximum likelihood estimators of the parameters for both procedures were determined. In addition, distribution sampling techniques were used to check if such asymptotic results apply to small samples. The results show a gain in precision on the estimates of the parameters when they are calculated using the proposed trivariate distribution and such a gain is more significant in relation to the parameters of shorter samples. A region located in Southern Mexico with 28 streamflow gauging stations, has been chosen to apply the trivariate distribution to flood frequency analysis. Regional at-site estimates are compared with those obtained by the Station-year and Index Flood methods. In general, it could be concluded that the proposed trivariate extreme value distribution offers a better alternative to solve flood frequency analysis problems.