A dynamical mixture model for unsupervised tail estimation without threshold selection
This paper deals with an unsupervised approach to tail estimation. The authors model all data, not only those belonging to the tail. They use a mixture model, one component of which is a generalized Pareto distribution and the other component has a density distribution with light tail, and the weight in the mixture is dynamic and determines when the generalized Pareto distribution component in the mixture is prevalent. Let \(X_{1},\ldots,X_{n}\) be nonnegative i.i.d. random variables with common probability density function \[ l(x)= \{[1- p(x;\theta)] f(x;\beta)+ p(x;\theta)g(x;\xi,\sigma)\}/ Z(\theta,\beta,\xi,\sigma), \] where \(g(x;\xi,\sigma)=\sigma^{-1} (1+\xi x/\sigma)^{-1/\xi-1}\) is the generalized Pareto distribution density, \(p(x;\theta)\) is the mixing function taking values in \((0,1]\), increasing in \(x\) and such that \(\lim_{x\to x_{\infty}}p(x;\theta)=1\) for all \(\theta\), \(f(x;\beta)\) is a density, and \(Z(\theta,\beta,\xi,\sigma)\) is the normalizing constant. The authors show how to choose \(f\) and \(p\) and present an algorithm for sampling from \(l\). Likelihood based inference for the model is discussed and a numerical procedure is described. The authors compare the proposed dynamic mixture methods with Dupuis' robust threshold approach in peak-over-threshold inference. Simulation results are presented that indicate that the new approach can be useful in unsupervised tail estimation, especially in heavy tailed situations and for small percentiles.
scientific article; zbMATH DE number 2059067