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

TLDR

Copulas, especially the Frank Archimedean family, have been effective for bivariate hydrologic modeling, but extending them to trivariate or higher dimensions is constrained by limited dependency structures. This study investigates a non‑Archimedean Plackett copula grounded in constant cross‑product ratio theory to enable trivariate stochastic analysis of extreme rainfall. The authors numerically identify the feasible Plackett parameter range, then apply the resulting trivariate copula to Indiana rainfall stations, computing conditional expectations of peak intensity, time to peak, and cumulative rainfall percentages. Results demonstrate that the Plackett copula performs well at both bivariate and trivariate levels, preserving lower‑level dependencies while offering flexibility for continuous variables in multivariate rainfall analysis.

Abstract

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.