Inference for extremal regression with dependent heavy-tailed data

Abstract

Nonparametric inference on tail conditional quantiles and their least squares analogs, expectiles, remains limited to i.i.d. data. Expectiles are themselves quan- tiles of a transformation of the underlying distribution. We develop a fully operational kernel-based inferential theory for extreme conditional quantiles and expectiles in the challenging framework of ↵-mixing, conditional heavy-tailed data whose tail index may vary with covariate values. This extreme value problem requires a dedicated treatment to deal with data sparsity in the far tail of the response, in addition to handling diffi culties inher- ent to mixing, smoothing, and sparsity associated to covariate localization. We prove the pointwise asymptotic normality of our estimators and obtain optimal rates of convergence reminiscent of those found in the i.i.d. regression setting, but which had not been estab- lished in the conditional extreme value literature so far. Our mathematical assumptions are satisfied in location-scale models with possible temporal misspecification, nonlinear regression models, and autoregressive models, among others. We propose full bias and variance reduction procedures, and simple but e↵ective data-based rules for selecting tun- ing hyperparameters. Our inference strategy is shown to perform well in finite samples and is showcased in applications to stock returns and tornado loss data.

Keywords

Reference

Abdelaati Daouia, Gilles Stupfler, and Antoine Usseglio-Carleve, “Inference for extremal regression with dependent heavy-tailed data”, Annals of Statistics, vol. 51, n. 5, December 2023, pp. 2040–2066.