Statistical Estimation of Extreme Values in the Mixture Distributions

In this paper, we address possible bias issues in quantile estimation using generalized extreme value distribution (GEV). We first provide two examples, one from a Fréchet-Gumbel mixture distribution and the other from a Gumbel-Gumbel mixture distribution, which explain theoretical asymptotic convergence of extreme value estimators to GEV in cases of mixture distributions. However, through a simple example, we show that the convergence rate can be arbitrarily slow for some cases, and explain that slow convergence rates can create bias in actual statistical estimations of the quantile of extreme values. To reduce the bias in quantile estimation of extreme values, we briefly mention that (modified) GPD method can be effective, depending on the data size available. Finally, actual data on the precipitation in Seoul are analyzed using both the GEV and GPD method.