A Discrete GEV Distribution

Annual maximum data are frequently recorded as integers, yet they are routinely modeled with the continuous generalized extreme value (GEV) distribution. We study the discrete GEV distribution of Dkengne et al. (2016), which is the distribution of the integer part of a continuous GEV random variable, and we provide a systematic treatment of its distributional, inferential and finite-sample properties. The model retains the location, scale and shape parameters of its continuous counterpart and contains the discrete Gumbel, discrete Fréchet and discrete (reversed) Weibull distributions as particular cases. We give the support (including the endpoint cases), the probability mass and cumulative distribution functions, their tail behavior, moments, the quantile function, and quantile-based measures of skewness and kurtosis. For maximum likelihood estimation, we identify the subsets of the parameter space on which the support is locally fixed, show that the model is differentiable in quadratic mean on a larger set that also contains integer endpoints when β>−12, and state the resulting asymptotic normality. We also show that flooring, ceiling and rounding of a GEV variable all lead to the same discrete family up to a shift in location. Simulation studies examine the finite-sample performance of the estimators and compare the discrete GEV with the continuous GEV fitted to the integer values under several observation mechanisms, including block maxima of integer counts. Applications to annual maximum wind speeds at Lisbon and annual maximum temperatures at Oxford are analyzed with discrete versions of the Kolmogorov–Smirnov, Cramér–von Mises and Anderson–Darling statistics calibrated by a parametric bootstrap.