A note on the applicability of log-Gumbel and log-logistic probability distributions in hydrological analyses: II. Assumed pdf

Abstract

Applicability of log-Gumbel (LG) and log-logistic (LL) probability distributions in hydrological studies is critically examined under real conditions, where the assumed distribution differs from the true one. The set of alternative distributions consists of five two-parameter distributions with zero lower bound, including LG and LL as well as lognormal (LN), linear diffusion analogy (LD) and gamma (Ga) distributions. The log-Gumbel distribution is considered as both a false and a true distribution. The model error of upper quantiles and of the first two moments is analytically derived for three estimation methods: the method of moments (MOM), the linear moments method (LMM) and the maximum likelihood method (MLM). These estimation methods are used as methods of approximation of one distribution by another distribution. As recommended in the first of this two-part series of papers, MLM turns out to be the worst method, if the assumed LG or LL distribution is not the true one. It produces a huge bias of upper quantiles, which is at least one order higher than that of the other two methods. However, the reverse case, i.e. acceptance of LN, LD or Ga as a hypothetical distribution, while the LG or LL distribution is the true one, gives the MLM bias of reasonable magnitude in upper quantiles. Therefore, one should avoid choosing the LG and LL distributions in flood frequency analysis, especially if MLM is to be applied.