Trivariate statistical analysis of extreme rainfall events via the Plackett family of copulas

Recent hydrologic studies on multivariate stochastic analysis have indicated that copulas perform well for bivariate problems. In particular, the Frank family of Archimedean copulas has been a popular choice for a dependence model. However, there are limitations to extending such Archimedean copulas to trivariate and higher dimensions, with very specific restrictions on the kinds of dependencies that can be modeled. In this study, we examine a non-Archimedean copula from the Plackett family that is founded on the theory of constant cross-product ratio. It is shown that the Plackett family not only performs well at the bivariate level, but also allows a trivariate stochastic analysis where the lower-level dependencies between variables can be fully preserved while allowing for specificity at the trivariate level as well. The feasible range of Plackett parameters that would result in valid 3-copulas is determined numerically. The trivariate Plackett family of copulas is then applied to the study of temporal distribution of extreme rainfall events for several stations in Indiana where the estimated parameters lie in the feasible region. On the basis of a given rainfall depth and duration, conditional expectations of rainfall features such as expected peak intensity, time to peak, and percentage cumulative rainfall at 10% cumulative time increments are evaluated. The results of this study suggest that while the constant cross-product ratio theory was conventionally applied to discrete type random variables, it is also applicable to continuous random variables, and that it provides further flexibility for multivariate stochastic analyses of rainfall.