Nonparametric extreme conditional expectile estimation

Econometrics and Statistics

2022

Self Cite

Quantiles and expectiles can be interpreted as solutions of convex minimization problems. Unlike quantiles, expectiles are determined by tail expectations rather than tail probabilities, and define a coherent risk measure. For these reasons, among others, they have recently been the subject of renewed attention in actuarial and financial risk management. Here, we focus on the challenging problem of estimating extreme expectiles, whose order converges to one as the sample size increases, given a functional covariate. We construct a functional kernel estimator of extreme conditional expectiles by writing expectiles as quantiles of a different distribution. The asymptotic properties of the estimators are studied in the context of conditional heavy-tailed distributions. We also provide and analyse different ways of estimating the functional tail index, as a way to extrapolate our estimates to the very far conditional tails. A numerical illustration of the finite-sample performance of our estimators is provided on simulated and real datasets.

“…We refer to Girard et al (2021) for a similar approach in the case where the covariate X is finite-dimensional.…”

Section: Notation and Assumptionsmentioning

confidence: 99%

Section: Extreme Functional Expectile Extrapolationmentioning

confidence: 99%

Section: Cross-validation Proceduresmentioning

confidence: 99%

Section: Real Data Examplementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Functional estimation of extreme conditional expectiles

Girard¹,

Econometrics and Statistics

2022

Self Cite

Composite bias‐reduced Lp‐quantile‐based estimators of extreme quantiles and expectiles

2022

Can J Statistics

Self Cite

Quantiles are a fundamental concept in extreme value theory. They can be obtained from a minimization framework using an absolute error loss criterion. The companion notion of expectiles, based on squared rather than absolute error loss minimization, has received substantial attention from the fields of actuarial science, finance and econometrics over the last decade. Quantiles and expectiles can be embedded in a common framework of L p −quantiles, whose extreme value properties have been explored very recently. Although this generalized notion of quantiles has shown potential for the estimation of extreme quantiles and expectiles, available estimators remain quite difficult to use: they suffer from substantial bias and the question of the choice of the tuning parameter p remains open. In this paper, we work in a context of heavy tails, and we construct composite bias-reduced estimators of extreme quantiles and expectiles based on L p −quantiles. We provide a discussion of the data-driven choice of p and of the anchor L p −quantile level in practice. The proposed methodology is compared to existing approaches on simulated data and real data.

Extreme $$L^p$$-quantile Kernel Regression

Girard

Advances in Contemporary Statistics and Econometrics

2021

Quantiles are recognized tools for risk management and can be seen as minimizers of an L 1 −loss function, but do not define coherent risk measures in general. Expectiles, meanwhile, are minimizers of an L 2 −loss function and define coherent risk measures; they have started to be considered as good alternatives to quantiles in insurance and finance. Quantiles and expectiles belong to the wider family of L p −quantiles. We propose here to construct kernel estimators of extreme conditional L p −quantiles. We study their asymptotic properties in the context of conditional heavy-tailed distributions and we show through a simulation study that taking p ∈ (1, 2) may allow to recover extreme conditional quantiles and expectiles accurately. Our estimators are also showcased on a real insurance data set.