Mixed estimation methods for Halphen distributions with applications in extreme hydrologic events

The Halphen family of distributions is a flexible and complete system to fit sets of observations independent and identically distributed. Recently, it is shown that this family of distributions represents a potential alternative to the generalized extreme value distributions to model extreme hydrological events. The existence of jointly sufficient statistics for parameter estimation leads to optimality of the method of maximum likelihood (ML). Nevertheless, the ML method requires numerical approximations leading to less accurate values. However, estimators by the method of moments (MM) are explicit and their computation is fast. Even though MM method leads to good results, it is not optimal. In order to combine the advantages of the ML (optimality) and MM (efficiency and fast computations), two new mixed methods were proposed in this paper. One of the two methods is direct and the other is iterative, denoted respectively direct mixed method (MMD) and iterative mixed method (MMI). An overall comparison of the four estimation methods (MM, ML, MMD and MMI) was performed using Monte Carlo simulations regarding the three Halphen distributions. Generally, the MMI method can be considered for the three Halphen distributions since it is recommended for a majority of cases encountered in hydrology. The principal idea of the mixed methods MMD and MMI could be generalized for other distributions with complicated density functions.