On a relationship between randomly and non-randomly thresholded empirical average excesses for heavy tails

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.

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“…Note now that a combination of the local uniformity of condition  2 (𝛾, 𝜌, A) (see e.g., Lemma 2 in Stupfler, 2019) with the assumption A((1…”

Section: Proofs Of Main Resultsmentioning

confidence: 99%

“…Now, by the local uniformity of condition  2 (𝛾, 𝜌, A) (see e.g., Lemma 2 in Stupfler, 2019) combined with the assumption A(1∕F…”

Section: Note That Fmentioning

confidence: 99%

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

Stupfler

Usseglio-Carleve

2022

Can J Statistics

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“…This is a modified version of the empirical upper tail copula in Equation ( 13) of Schmidt and Stadtmüller (2006). Adapting Lemma 7 from Stupfler (2019) shows that it is a locally uniformly consistent estimator of R j, on (0, ∞) 2 under our technical conditions. Combining these tools, we arrive at the estimators…”

Section: Optimal Choices Of Weightsmentioning

confidence: 88%

Optimal pooling and distributed inference for the tail index and extreme quantiles

Daouia¹,

Padoan²,

Stupfler³

2021

Preprint

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This paper investigates pooling strategies for tail index and extreme quantile estimation from heavy-tailed data. To fully exploit the information contained in several samples, we present general weighted pooled Hill estimators of the tail index and weighted pooled Weissman estimators of extreme quantiles calculated through a nonstandard geometric averaging scheme. We develop their large-sample asymptotic theory across a fixed number of samples, covering the general framework of heterogeneous sample sizes with different and asymptotically dependent distributions. Our results include optimal choices of pooling weights based on asymptotic variance and MSE minimization. In the important application of distributed inference, we prove that the variance-optimal distributed estimators are asymptotically equivalent to the benchmark Hill and Weissman estimators based on the unfeasible combination of subsamples, while the AMSE-optimal distributed estimators enjoy a smaller AMSE than the benchmarks in the case of large bias. We consider additional scenarios where the number of subsamples grows with the total sample size and effective subsample sizes can be low. We extend our methodology to handle serial dependence and the presence of covariates. Simulations confirm that our pooled estimators perform virtually as well as the benchmark estimators. Two applications to real weather and insurance data are showcased.

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“…Our next result, of interest in its own right, examines the joint convergence between Hill estimators and intermediate order statistics across marginals. A related result, limited to joint convergence of Hill estimators only, is Theorem 4 in Stupfler (2019).…”

Section: At the Intermediate Levelmentioning

confidence: 99%

Joint inference on extreme expectiles for multivariate heavy-tailed distributions

Padoan¹,

Stupfler²

2020

Preprint

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The notion of expectiles, originally introduced in the context of testing for homoscedasticity and conditional symmetry of the error distribution in linear regression, induces a law-invariant, coherent and elicitable risk measure that has received a significant amount of attention in actuarial and financial risk management contexts. A number of recent papers have focused on the behaviour and estimation of extreme expectile-based risk measures and their potential for risk management. Joint inference of several extreme expectiles has however been left untouched; in fact, even the inference of a marginal extreme expectile turns out to be a difficult problem in finite samples. We investigate the simultaneous estimation of several extreme marginal expectiles of a random vector with heavy-tailed marginal distributions. This is done in a general extremal dependence model where the emphasis is on pairwise dependence between the margins. We use our results to derive accurate confidence regions for extreme expectiles, as well as a test for the equality of several extreme expectiles. Our methods are showcased in a finite-sample simulation study and on real financial data.

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On a relationship between randomly and non-randomly thresholded empirical average excesses for heavy tails

Cited by 8 publications

References 55 publications

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

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

Optimal pooling and distributed inference for the tail index and extreme quantiles

Joint inference on extreme expectiles for multivariate heavy-tailed distributions

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