A Data-Retaining Framework for Tail Estimation

Modeling of extreme data often involves thresholding, or retaining only the most extreme observations, in order that the tail may "speak" and not be overwhelmed by the bulk of the data. We describe a transformation-based framework that allows univariate density estimation to smoothly transition from a flexible, semi-parametric estimation of the bulk into a parametric estimation of the tail without thresholding. In the limit, this framework has desirable theoretical tail-matching properties to the selected parametric distribution. We develop three Bayesian models under the framework: one using a logistic Gaussian process (LGP) approach; one using a Dirichlet process mixture model (DPMM); and one using a predictive recursion approximation of the DPMM. Models produce estimates and intervals for density, distribution, and quantile functions across the full data range and for the tail index (inverse-power-decay parameter), under an assumption of heavy tails. For each approach, we carry out a simulation study to explore the model's practical usage in non-asymptotic settings, comparing its performance to methods that involve thresholding.

Among the three models proposed, the LGP has lowest bias through the bulk and highest quantile interval coverage generally. Compared to thresholding methods, its tail predictions have lower root mean squared error (RMSE) in all scenarios but the most complicated, e.g. a sharp bulk-to-tail transition. The LGP's consistent underestimation of the tail index does not hinder tail estimation in pre-extrapolation to moderate-extrapolation regions but does affect extreme extrapolations.

An interplay between the parametric transform and the natural sparsity of the DPMM sometimes causes the DPMM to favor estimation of the bulk over estimation of the tail. This can be overcome by increasing prior precision on less sparse (flatter) base-measure density shapes. A finite mixture model (FMM), substituted for the DPMM in simulation, proves effective at reducing tail RMSE over thresholding methods in some, but not all, scenarios and quantile levels.

The predictive recursion marginal posterior (PRMP) model is fast and does the best job among proposed models of estimating the tail-index parameter. This allows it to reduce RMSE in extrapolation over thresholding methods in most scenarios considered. However, bias from the predictive recursion contaminates the tail, casting doubt on the PRMP's predictions in tail regions where data should still inform estimation. We recommend the PRMP model as a quick tool for visualizing the marginal posterior over transformation parameters, which can aid in diagnosing multimodality and informing the precision needed to overcome sparsity in the mixture model approach.

In summary, there is not enough information in the likelihood alone to prevent the bulk from overwhelming the tail. However, a model that harnesses the likelihood with a carefully specified prior can allow both the bulk and tail to speak without an explicit separation of the two. Moreover, retaining all of the data under this framework reduces quantile variability, improving prediction in the tails compared to methods that threshold.