{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:24:54Z","timestamp":1760145894605,"version":"build-2065373602"},"reference-count":37,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2024,9,18]],"date-time":"2024-09-18T00:00:00Z","timestamp":1726617600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004242","name":"Princess Nourah bint Abdulrahman University","doi-asserted-by":"publisher","award":["PNURSP2024R515","R.G.P. 1\/128\/45"],"award-info":[{"award-number":["PNURSP2024R515","R.G.P. 1\/128\/45"]}],"id":[{"id":"10.13039\/501100004242","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Deanship of Scientific Research and Graduate Studies at King Khalid University","award":["PNURSP2024R515","R.G.P. 1\/128\/45"],"award-info":[{"award-number":["PNURSP2024R515","R.G.P. 1\/128\/45"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This paper treats the problem of risk management through a new conditional expected shortfall function. The new risk metric is defined by the expectile as the shortfall threshold. A nonparametric estimator based on the Nadaraya\u2013Watson approach is constructed. The asymptotic property of the constructed estimator is established using a functional time-series structure. We adopt some concentration inequalities to fit this complex structure and to precisely determine the convergence rate of the estimator. The easy implantation of the new risk metric is shown through real and simulated data. Specifically, we show the feasibility of the new model as a risk tool by examining its sensitivity to the fluctuation in financial time-series data. Finally, a comparative study between the new shortfall and the standard one is conducted using real data.<\/jats:p>","DOI":"10.3390\/e26090798","type":"journal-article","created":{"date-parts":[[2024,9,19]],"date-time":"2024-09-19T04:59:54Z","timestamp":1726721994000},"page":"798","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Nonparametric Expectile Shortfall Regression for Complex Functional Structure"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0009-0009-3860-6917","authenticated-orcid":false,"given":"Mohammed B.","family":"Alamari","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, Abha 62529, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1306-7688","authenticated-orcid":false,"given":"Fatimah A.","family":"Almulhim","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, College of Sciences, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9684-1589","authenticated-orcid":false,"given":"Zoulikha","family":"Kaid","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, Abha 62529, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6527-5783","authenticated-orcid":false,"given":"Ali","family":"Laksaci","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, Abha 62529, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,9,18]]},"reference":[{"key":"ref_1","first-page":"819","article-title":"Asymmetric least squares estimation and testing","volume":"55","author":"Newey","year":"1987","journal-title":"Econom. J. Econom. Soc."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1177\/1471082X14561155","article-title":"Expectile and quantile regression\u2014David and Goliath","volume":"15","author":"Waltrup","year":"2015","journal-title":"Stat. Model."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"487","DOI":"10.1080\/1351847X.2015.1052150","article-title":"Risk management with expectiles","volume":"23","author":"Bellini","year":"2017","journal-title":"Eur. J. Financ."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1007\/s10260-018-00434-w","article-title":"Backtesting VaR and expectiles with realized scores","volume":"28","author":"Bellini","year":"2019","journal-title":"Stat. Methods Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1007\/s10994-018-5762-9","article-title":"Learning rates for kernel-based expectile regression","volume":"108","author":"Farooq","year":"2019","journal-title":"Mach. Learn."},{"key":"ref_6","unstructured":"Chakroborty, S., Iyer, R., and Trindade, A.A. (2024). On the use of the M-quantiles for outlier detection in multivariate data. arXiv."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2661","DOI":"10.1214\/15-AOS1431","article-title":"High-dimensional generalizations of asymmetric least squares regression and their applications","volume":"44","author":"Gu","year":"2016","journal-title":"Ann. Stat."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"304","DOI":"10.1016\/j.spl.2018.02.006","article-title":"Expectile regression for analyzing heteroscedasticity in high dimension","volume":"137","author":"Zhao","year":"2018","journal-title":"Stat. Probab. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1177\/1471082X13494159","article-title":"Beyond mean regression","volume":"13","author":"Kneib","year":"2013","journal-title":"Stat. Model."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"104673","DOI":"10.1016\/j.jmva.2020.104673","article-title":"The consistency and asymptotic normality of the kernel type expectile regression estimator for functional data","volume":"181","author":"Mohammedi","year":"2021","journal-title":"J. Multivar. Anal."},{"key":"ref_11","first-page":"131","article-title":"Functional estimation of extreme conditional expectiles","volume":"21","author":"Girard","year":"2022","journal-title":"Econom. Stat."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1111\/1467-9965.00068","article-title":"Coherent measures of risk","volume":"9","author":"Artzner","year":"1999","journal-title":"Math. Financ."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1016\/j.jeconbus.2014.11.002","article-title":"A comparison of expected shortfall estimation models","volume":"78","author":"Righi","year":"2015","journal-title":"J. Econ. Bus."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"100391","DOI":"10.1016\/j.jcomm.2024.100391","article-title":"On the estimation of Value-at-Risk and Expected Shortfall at extreme levels","volume":"34","author":"Lazar","year":"2024","journal-title":"J. Commod. Mark."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"106151","DOI":"10.1016\/j.jspi.2024.106151","article-title":"A new non-parametric estimation of the expected shortfall for dependent financial losses","volume":"232","author":"Moutanabbir","year":"2024","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_16","first-page":"115","article-title":"Nonparametric estimation and sensitivity analysis of expected shortfall","volume":"14","author":"Scaillet","year":"2004","journal-title":"Math. Financ. Int. J. Math. Stat. Financ. Econ."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1016\/j.jeconom.2008.09.005","article-title":"Nonparametric estimation of conditional VaR and expected shortfall","volume":"147","author":"Cai","year":"2008","journal-title":"J. Econom."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"544","DOI":"10.1080\/00949655.2021.1966791","article-title":"Nonparametric estimation of expected shortfall via Bahadur-type representation and Berry\u2013Ess\u00e9en bounds","volume":"92","author":"Wu","year":"2022","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1080\/07474938.2013.807107","article-title":"Conditional VAR and expected shortfall: A new functional approach","volume":"35","author":"Ferraty","year":"2016","journal-title":"Econom. Rev."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Ait-Hennani, L., Kaid, Z., Laksaci, A., and Rachdi, M. (2022). Nonparametric estimation of the expected shortfall regression for quasi-associated functional data. Mathematics, 10.","DOI":"10.3390\/math10234508"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Fuchs, S., Schlotter, R., and Schmidt, K.D. (2017). A review and some complements on quantile risk measures and their domain. Risks, 5.","DOI":"10.3390\/risks5040059"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Almanjahie, I.M., Bouzebda, S., Kaid, Z., and Laksaci, A. (2024). The local linear functional kNN estimator of the conditional expectile: Uniform consistency in number of neighbors. Metrika, Springer.","DOI":"10.1007\/s00184-023-00942-0"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"105281","DOI":"10.1016\/j.jmva.2023.105281","article-title":"Asymptotic normality of the local linear estimator of the functional expectile regression","volume":"202","author":"Litimein","year":"2024","journal-title":"J. Multivar. Anal."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/j.jmva.2018.11.007","article-title":"Recent advances in functional data analysis and high-dimensional statistics","volume":"170","author":"Aneiros","year":"2019","journal-title":"J. Multivar. Anal."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jmva.2015.12.001","article-title":"An introduction to recent advances in high\/infinite dimensional statistics [Editorial]","volume":"170","author":"Goia","year":"2016","journal-title":"J. Multivar. Anal."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Yu, D., Pietrosanu, M., Mizera, I., Jiang, B., Kong, L., and Tu, W. (2024). Functional Linear Partial Quantile Regression with Guaranteed Convergence for Neuroimaging Data Analysis. Statistics in Biosciences, Springer.","DOI":"10.1007\/s12561-023-09412-7"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"105292","DOI":"10.1016\/j.jmva.2023.105292","article-title":"Estimation of extreme multivariate expectiles with functional covariates","volume":"202","author":"Laloe","year":"2024","journal-title":"J. Multivar. Anal."},{"key":"ref_28","first-page":"71","article-title":"Nonlinear autoregressive pro cesses","volume":"360","author":"Jones","year":"1978","journal-title":"Proc. R. Soc. Lond. A Math. Phys. Sci."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"84","DOI":"10.2307\/3212926","article-title":"Non-linear time series models for non-linear random vibrations","volume":"17","author":"Ozaki","year":"1980","journal-title":"J. Appl. Prob."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"987","DOI":"10.2307\/1912773","article-title":"Autoregressive conditional heteroskedasticity with estimates of the variance of U.K. inflation","volume":"50","author":"Engle","year":"1982","journal-title":"Econometrica"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1016\/0304-4076(86)90063-1","article-title":"Generalized autoregressive conditional heteroskedasticity","volume":"31","author":"Bollerslev","year":"1986","journal-title":"J. Econom."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1016\/S0169-7161(01)19019-X","article-title":"Gaussian processes: Inequalities, small ball probabilities and applications","volume":"19","author":"Li","year":"2001","journal-title":"Handb. Stat."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1111\/rssb.12076","article-title":"Dynamic functional principal components","volume":"77","author":"Hallin","year":"2015","journal-title":"J. R. Stat. Soc. Ser. Stat. Methodol."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1289","DOI":"10.1080\/09603100701630030","article-title":"Empirical distributions of stock returns: Paris stock market, 1980\u20132003","volume":"16","author":"Kanellopoulou","year":"2008","journal-title":"Appl. Financ. Econ."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1080\/02664763.2016.1161738","article-title":"Real-time monitoring of carbon monoxide using value-at-risk measure and control charting","volume":"44","author":"Bersimis","year":"2017","journal-title":"J. Appl. Stat."},{"key":"ref_36","unstructured":"Ferraty, F., and Vieu, P. (2006). Nonparametric Functional Data Analysis: Theory and Practice, Springer."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"254","DOI":"10.1109\/TIT.1986.1057163","article-title":"Recursive probability density estimation for weakly dependent stationary processes","volume":"32","author":"Masry","year":"1986","journal-title":"IEEE Trans. Inf. 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