Generalized Truncated Fr閏het Generated Family Distributions and Their Applications

Abstract

Understanding a phenomenon from observed data requires contextual and efficient statistical models. Such models are based on probability distributions having sufficiently flexible statistical properties to adapt to a maximum of situations. Modern examples include the distributions of the truncated Fréchet generated family. In this paper, we go even further by introducing a more general family, based on a truncated version of the generalized Fréchet distribution. This generalization involves a new shape parameter modulating to the extreme some central and dispersion parameters, as well as the skewness and weight of the tails. We also investigate the main functions of the new family, stress-strength parameter, diverse functional series expansions, incomplete moments, various entropy measures, theoretical and practical parameters estimation, bivariate extensions through the use of copulas, and the estimation of the model parameters. By considering a special member of the family having the Weibull distribution as the parent, we fit two data sets of interest, one about waiting times and the other about precipitation. Solid statistical criteria attest that the proposed model is superior over other extended Weibull models, including the one derived to the former truncated Fréchet generated family.