Time‐change models for asymmetric processes

Abstract

AbstractMany records in environmental sciences exhibit asymmetric trajectories. The physical mechanisms behind these records may lead for example to sample paths with different characteristics at high and low levels (up–down asymmetries) or in the ascending and descending phases leading to time irreversibility (front–back asymmetries). Such features are important for many applications, and there is a need for simple and tractable models that can reproduce them. In this paper, we explore original time‐change models where the clock is a stochastic process that depends on the observed trajectory. The ergodicity of the proposed model is established under general conditions, and this result is used to develop nonparametric estimation procedures based on the joint distribution of the process and its derivative. The methodology is illustrated on meteorological and oceanographic data sets. We show that, combined with a marginal transformation, the proposed methodology is able to reproduce important characteristics of the data set such as marginal distributions, up‐crossing intensity, and up–down and front–back asymmetries.