The tapered (or generalized) Pareto distribution, also called the modified Gutenberg–Richter law, has been used to model the sizes of earthquakes. Unfortunately, maximum likelihood estimates of the cutoff parameter are substantially biased. Alternative estimates for the cutoff parameter are presented, and their properties discussed.

There is often interest in understanding how the extremely high/low values of different processes are related to each other. One possible way to tackle this problem is via an asymptotic approach, which involves fitting the multivariate generalized Pareto distribution (MGPD) to data that exceed a suitably high threshold. There are two possible definitions of the MGPD. The first, which has classi...

This paper studies recurrence properties of autoregressive (AR) processes with “super-heavy tailed” innovations. Specifically, we study the case where the innovations are distributed, roughly speaking, as log-Pareto random variables (i.e., the tail decay is essentially a logarithm raised to some power). We show that these processes exhibit interesting and somewhat surprising behavior. In partic...

Attempts to formulate mathematically precise definitions of basic concepts such as causality, randomness, and probability, have a long history. The concept of generalized definability that is outlined in this project suggests that such definitions may not exist. Furthermore, it suggests that existing definitions of many basic concepts, among them those of stability, statistical independence, an...

The Generalized Pareto (GP) and Generalized extreme value (GEV) distributions play an important role in extreme value analyses as models for threshold excesses and block maxima, respectively. For each of these distributions we consider Bayesian inference using “reference” prior distributions (in the general sense of priors constructed using formal rules) for the model parameters, specifically a...