Halphen Distribution System. I: Mathematical and Statistical Properties

In the 1940s, Étienne Halphen, a French statistician and hydrologist, developed a set of distributions for frequency analysis of river flows. Halphen's research was fueled by an extensive practical experience with hydrological data combined with a solid knowledge of statistics. Because of their complex form involving Bessel functions and exponential factorial functions, Halphen's distributions have remained for several years in oblivion. This paper revisits the three types of Halphen distributions, in particular their mathematical and statistical properties. Their flexible shapes and tail properties should make them excellent candidates for frequency analysis of extremes. A companion paper presents procedures for estimating parameters and quantiles of Halphen's distribution.