Extremal behavior of solutions to a stochastic difference equation, with applications to ARCH processes

AbstractWe consider limit distributions of extremes of a process {Yn} satisfying the stochastic difference equation Yn-AnYn−1+Bn, n⩾1,Y0⩾0, where {An, Bn} are i.i.d. R2+-valued random pairs, A special case of interest is when {Yn} is derived from a first order ARCH process. Parameters of the limit law are exhibited; some are hard to calculate explicitly but easy to simulate

Peaks Over Thresholds Modeling With Multivariate Generalized Pareto Distributions

When assessing the impact of extreme events, it is often not just a single component, but the combined behavior of several components which is important. Statistical modeling using multivariate generalized Pareto (GP) distributions constitutes the multivariate analogue of univariate peaks over thresholds modeling, which is widely used in finance and engineering. We develop general methods for construction of multivariate GP distributions and use them to create a variety of new statistical models. A censored likelihood procedure is proposed to make inference on these models, together with a threshold selection procedure, goodness-of-fit diagnostics, and a computationally tractable strategy for model selection. The models are fitted to returns of stock prices of four UK-based banks and to rainfall data in the context of landslide risk estimation. Supplementary materials and codes are available online