 CE40 - Mathématiques

# Extremal Regression with Applications to Econometrics, Environment and Finance – ExtremReg

## Extremal Regression with Applications to Econometrics, Environment and Finance

First, the investigation of mathematical properties under frontier models is often a lot harder than under the standard regression models. Second, while the extreme value properties of quantiles are well developed, much less is known about their expectile and extremile analogues. Third, the problem of estimating conditional extremes and risk measures in high dimensions has been much less discussed in the literature. Our project tries to solve <br />these issues in different directions.

### Frontier modeling, Extremal regression and Tail risk handling

This project concentrates around three themes that are central to the area of modern extreme value statistics. First, we contribute to the expanding literature on frontier modeling in three different directions:<br /><br />Task 1: Polynomial spline fitting of the frontier function under shape constraints.<br /><br />Task 2: Robust frontier estimation from noisy data.<br /><br />Task 3: Estimation of the parameters of locally stationary, one-sided autoregressive processes.<br /><br />Second, we further investigate the recent extreme value theory built on the use of expectiles and extremiles that have gained increasing interest in risk management:<br /><br />Task 4: Construct extrapolated estimators of tail extremiles and expectiles, and obtain their asymptotic distributions under mixing conditions. <br /><br />Task 5: Explore the statistical notion of expectile depth as well as expectile regression methods with multivariate response variables.<br /><br />Finally, we focus on the problem of estimating extremes and risk measures in the presence of covariates:<br /><br />Task 6: Provide a general toolbox for estimating tail regression expectiles and extremiles. <br /><br />Task 7: Use of the theory of multivariate extreme values ??in the regression framework, measuring the variability in the distribution tail, and comparing risks of different types.

Here we briefly summarize the methods utilized to achieve some objectives:
The main tool in our robust frontier analysis is a concept of partial regression boundary defined as a special probability-weighted moment. When it comes to estimating extreme risks for heavy tailed distributions, one way is to first estimate the tail index, e.g., via weighted combinations of top order statistics and asymmetric least squares estimates, before using the resulting estimator in conjunction with extrapolation and plug-in techniques as the basis for estimating the tail risk measures. Extremile regression relies on a local linear approach for estimating central conditional extremiles and on the extrapolation of a nonparametric quantile estimator at the far tails. Extreme expectile regression exploits the characterization of expectiles as quantiles of a transformation of the initial distribution to define original kernel estimators. To remedy the curse of dimensionality when the number of covariates becomes too large, alternative estimators are constructed by making use of the regression residuals in conjunction with extreme value arguments. When considering bias-corrected estimation of the stable tail dependence function in the regression context, we first estimate the bias of a smoothed estimator of the tail dependence function, and then we subtract it from the estimator. A divergence criterion is employed to achieve robustness in the estimation of the tail dependence coefficient in presence of covariates. The construction of accurate confidence intervals for intermediate conditional quantiles is based on the combination of a nearest neighbor selection method and a dimension reduction technique. Finally, the new measure of variability in the distribution tail is obtained by applying a Box-Cox transformation to the Tail-Gini functional.

We present a robust approach for frontier estimation in deterministic models and a test of separability of unobservables in endogenous models. We estimate extreme expectiles by using a new ALS tail index estimator or by relying on L^p-quantiles to reduce the estimation bias. We also propose an automatic construction of reduced-bias estimators and solve the problem of
inference on extreme expectile-based risk measures in a general ß-mixing framework, as well as in a multivariate dependence context. The R package “ExtremeRisks” implements these procedures. We explore the notions of expectile depth and multivariate expectiles, as well as extremile regression. We define kernel estimators of extreme conditional expectiles and generalize
the approach to the class of L^p-quantiles. We solve the problem of estimating extreme expectiles in heteroscedastic regression models, as well as in the presence of functional covariates. We revisit the estimation of extreme quantiles in location-dispersion regression models. We construct robust confidence intervals to the dimension of the covariate for intermediate quantiles. We discuss the problems of bias reduction and robustness when estimating the tail dependence function in the regression framework. We introduce an intermediate estimator of the “Marginal Expected Shortfall” in the presence of covariates before extrapolating it beyond the sample, and we adapt this method to the estimation of reinsurance premiums when the amount of claims is observed jointly with covariates. We also consider the estimation of the «Expected Proportional Shortfall« which allows to compare risks of different nature, and we suggest a new measure of variability in the distribution tail.

We currently focus on the estimation of monotonic boundaries through cubic spline fitting and from noisy data. We are also developping the asymptotic theory of empirical extremiles under conditions of weak dependence. A project in the process of being completed investigates extremile regression for heavy-tailed distributions with application to seismic data. Another work under progress is concerned with the extension of the notion of expectile depth to the general class of M-quantiles which encompasses both expectiles and ordinary quantiles. In a different project, we try to improve the estimation of extreme
expectiles in heteroscedastic regression models by developing a more general theory in conjunction with a local linear approach. The construction of confidence intervals for extreme conditional quantiles, which remain accurate even when the dimension of the covariate increases, is also in full development. Moreover, we are working on two questions of utmost interest related to the robust estimation of the tail dependence function, not considered so far in the literature as it is a non-trivial problem, and to the design of a new risk measure parametrized by an index whose different values ??allow to estimate several important measures in actuarial science. We plan to investigate these problems in the regression case.

12 publications + 1 R package :

1. Ag Ahmad, A., Deme, E., Diop, A., Girard, S., Usseglio-Carleve, A. (2020). Estimation of extreme quantiles of heavy-tailed distributions in a location-dispersion regression model. Electronic Journal of Statistics, 14(2):4421-4456.
2. Daouia, A., Florens, J-P., Simar, L. (2020). Robustified expected maximum production frontiers, Econometric Theory, To appear.
3. Daouia, A., Gijbels, I., Stupfler, G. (2021). Extremile regression, Journal of the American Statistical Association, To appear.
4. Daouia, A., Girard, S., Stupfler, G. (2021). ExpectHill estimation, extreme risk and heavy tails, Journal of Econometrics 221(1): 97-117.
5. Escobar-Bach, M., Guillou, A., Goegebeur, Y. (2020). Bias correction in conditional multivariate extremes, Electron. J. Statist., 14, 1773-1795.
6. Gardes, L. (2020). Nonparametric confidence interval for conditional quantiles with large-dimensional covariates, Electronic Journal of Statistics, 14(1), 661-701.
7. Gardes, L., Girard, S. (2021). On the estimation of the variability in the distribution tail, Test, To appear.
8. Girard, S., Stupfler, G., Usseglio-Carleve, A. (2020). Nonparametric extreme conditional expectile estimation, Scandinavian Journal of Statistics, To appear.
9. Girard, S., Stupfler, G., Usseglio-Carleve, A. (2021). Extreme Lp-quantile kernel regression. To appear in Advances in Contemporary Statistics and Econometrics, Springer.
10. Goegebeur, Y., Guillou, A., Le Ho, N. K., Qin, J. (2020). Robust nonparametric estimation of the conditional tail dependence coefficient, J. Multivariate Anal., 178, To appear.
11. Goegebeur, Y., Guillou, A., Le Ho, N. K., Qin, J. (2021). Conditional marginal expected shortfall, Extremes, To appear.
12. Goegebeur, Y., Guillou, A., Qin, J. (2021). Extreme value estimation of the conditional risk premium in reinsurance, Insurance Math. Econom., 96, 68-80.
13. Padoan, S.A., Stupfler, G. (2020). R package « ExtremeRisks » : Extreme Risk Measures.

The project concentrates around three themes that are central to the area of modern extreme value statistics. First, we contribute to the expanding literature on non regular regression models where the observation errors are assumed to be one-sided and the regression function describes some frontier or boundary curve. This is motivated from abundant applications especially in production econometrics. The development of mathematical properties under these frontier models is, however, often a lot harder than under the standard regression models. In particular, classical regularity conditions are violated, hence why these models are typically referred to as non regular. Our project tries to solve these difficulties in different directions, namely polynomial spline fitting under shape constraints, estimation from noisy data using inverse problems, and estimation of locally stationary, one-sided autoregressive processes.

Second, we further investigate the recent extreme value theory built on the use of asymmetric least squares instead of order statistics. We focus on two least squares analogues of quantiles, called expectiles and extremiles. These concepts have gained increasing interest in risk management. While the extreme value properties of expectiles are well developed, much less is known about tail extremiles. For this reason, we aim to establish weighted approximations of the tail empirical extremile process, valid under mixing conditions. This part of the project is also dedicated to statistical expectile depth and multiple-output expectile regression methods.

Finally, we explore the important problem of estimating conditional and joint extremes in high dimension, which is still in full development.
The objectives of this part of the project are twofold. On the one hand, we will provide a general toolbox for estimating tail regression expectiles and extremiles in high dimension. On the other hand, we will discuss dimension reduction techniques when modeling the joint extremes of simultaneous time series.

## Project coordinator

Monsieur Abdelaati Daouia (FONDATION JEAN JACQUES LAFFONT)

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.

## Partner

IRMA_UNISTRA Institut de recherche mathématique avancée (UMR 7501)
INRIA GRA Centre de Recherche Inria Grenoble - Rhône-Alpes
TSE FONDATION JEAN JACQUES LAFFONT
Université Libre de Bruxelles / European Center for Advanced Research in Economics and Statistics (ECARES)
Seoul National University / Department of Statistics
Technische Universität Braunschweig / Institut für Mathematische Stochastik
ENSAI ENSAI Rennes & CREST

Help of the ANR 159,000 euros
Beginning and duration of the scientific project: September 2019 - 48 Months