{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:27:17Z","timestamp":1760146037348,"version":"build-2065373602"},"reference-count":45,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2024,9,30]],"date-time":"2024-09-30T00:00:00Z","timestamp":1727654400000},"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 Researchers","doi-asserted-by":"publisher","award":["PNURSP2024R515","R.G.P.\/128\/45"],"award-info":[{"award-number":["PNURSP2024R515","R.G.P.\/128\/45"]}],"id":[{"id":"10.13039\/501100004242","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia","award":["PNURSP2024R515","R.G.P.\/128\/45"],"award-info":[{"award-number":["PNURSP2024R515","R.G.P.\/128\/45"]}]},{"name":"Deanship of Scientific Research and Graduate Studies at King Khalid University","award":["PNURSP2024R515","R.G.P.\/128\/45"],"award-info":[{"award-number":["PNURSP2024R515","R.G.P.\/128\/45"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The main aim of this paper is to consider a new risk metric that permits taking into account the spatial interactions of data. The considered risk metric explores the spatial tail-expectation of the data. Indeed, it is obtained by combining the ideas of expected shortfall regression with an expectile risk model. A spatio-functional Nadaraya\u2013Watson estimator of the studied metric risk is constructed. The main asymptotic results of this work are the establishment of almost complete convergence under a mixed spatial structure. The claimed asymptotic result is obtained under standard assumptions covering the double functionality of the model as well as the data. The impact of the spatial interaction of the data in the proposed risk metric is evaluated using simulated data. A real experiment was conducted to measure the feasibility of the Spatio-Functional Expectile Shortfall Regression (SFESR) in practice.<\/jats:p>","DOI":"10.3390\/axioms13100678","type":"journal-article","created":{"date-parts":[[2024,9,30]],"date-time":"2024-09-30T12:06:32Z","timestamp":1727697992000},"page":"678","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Spatio-Functional Nadaraya\u2013Watson Estimator of the Expectile Shortfall Regression"],"prefix":"10.3390","volume":"13","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\/0000-0001-9198-4903","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":[[2024,9,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Cressie, N.A. (1993). Statistics for Spatial Data, Wiley.","DOI":"10.1002\/9781119115151"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Diggle, P., and Ribeiro, P.J. (2007). Model-Based Geostatistics, Springer.","DOI":"10.1007\/978-0-387-48536-2"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/0047-259X(90)90059-Q","article-title":"Kernel density estimation on random fields","volume":"34","author":"Tran","year":"1990","journal-title":"J. Multivar. Anal."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1016\/j.spl.2003.08.014","article-title":"Spatial kernel regression: Weak consistency","volume":"68","author":"Lu","year":"2004","journal-title":"Statist. Probab. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1023\/B:SISP.0000049116.23705.88","article-title":"Nonparametric spatial prediction","volume":"7","author":"Biau","year":"2004","journal-title":"Stat. Inference Stoch. Process."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/S0378-3758(02)00507-4","article-title":"Nonparametric kernel estimation of an isotropic variogram","volume":"121","year":"2004","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2469","DOI":"10.1214\/009053604000000850","article-title":"Local linear spatial regression","volume":"32","author":"Hallin","year":"2004","journal-title":"Ann. Stat."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"778","DOI":"10.1016\/j.jspi.2006.06.008","article-title":"Kernel regression estimation for random fields","volume":"137","author":"Carbon","year":"2007","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1080\/10485250801976717","article-title":"L1-estimation for spatial nonparametric regression","volume":"20","author":"Xu","year":"2008","journal-title":"J. Nonparametr. Stat."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1016\/j.jspi.2008.04.023","article-title":"Nonparametric estimation of conditional expectation","volume":"139","author":"Li","year":"2009","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"298","DOI":"10.3103\/S1066530707040023","article-title":"Kernel regression estimation for continuous spatial processes","volume":"16","author":"Yao","year":"2007","journal-title":"Math. Meth. Stat."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1075","DOI":"10.1016\/j.crma.2009.06.012","article-title":"Estimation non param\u00e9trique de quantiles conditionnels pour des variables fonctionnelles spatialement d\u00e9pendantes","volume":"347","author":"Laksaci","year":"2009","journal-title":"C. R. Math."},{"key":"ref_13","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_14","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_15","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, 1\u201329.","DOI":"10.1007\/s00184-023-00942-0"},{"key":"ref_16","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_17","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_18","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_19","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_20","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_21","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_22","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_23","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_24","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_25","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_26","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_27","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_28","first-page":"93","article-title":"Regression percentiles using asymmetric squared error loss","volume":"1","author":"Efron","year":"1991","journal-title":"Stat. Sin."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"755","DOI":"10.1016\/j.csda.2010.11.015","article-title":"Geoadditive expectile regression","volume":"56","author":"Sobotka","year":"2012","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/j.neucom.2017.03.040","article-title":"Expectile regression neural network model with applications","volume":"247","author":"Jiang","year":"2017","journal-title":"Neurocomputing"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1111\/rssb.12254","article-title":"Estimation of tail risk based on extreme expectiles","volume":"80","author":"Daouia","year":"2018","journal-title":"J. R. Stat. Soc. Ser. B Stat. Methodol."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1515\/demo-2017-0002","article-title":"Multivariate extensions of expectiles risk measures","volume":"5","author":"Said","year":"2017","journal-title":"Depend. Model."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Maume-Deschamps, V., Rulli\u00e8re, D., and Said, K. (2018). Asymptotics multivariate expectiles. arXiv.","DOI":"10.1515\/strm-2017-0014"},{"key":"ref_34","first-page":"131","article-title":"Functional estimation of extreme conditional expectiles","volume":"21","author":"Girard","year":"2022","journal-title":"Econom. Stat."},{"key":"ref_35","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","volume":"170","author":"Goia","year":"2016","journal-title":"J. Multivar. Anal."},{"key":"ref_36","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. Stat. Biosci., 1\u201317.","DOI":"10.1007\/s12561-023-09412-7"},{"key":"ref_37","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_38","unstructured":"Guyon, X. (1987, January 14\u201315). Estimation d\u2019un champ par pseudo-vraisemblance conditionnelle: Etude asymptotique et application au cas Markovien. Proceedings of the Sixth Franco-Belgian Meeting of Statisticians, Bruxelles, Belguim."},{"key":"ref_39","unstructured":"Ferraty, F., and Vieu, P. (2006). Nonparametric Functional Data Analysis: Theory and Practice, Springer. Springer Series in Statistics."},{"key":"ref_40","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":"Hanbook Stat."},{"key":"ref_41","first-page":"9","article-title":"Estimation of the density and the regression function under mixing conditions","volume":"19","author":"Liebscher","year":"2001","journal-title":"Stat. Decis."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"627","DOI":"10.1007\/s00184-021-00846-x","article-title":"Expectile regression for spatial functional data analysis (sFDA)","volume":"85","author":"Rachdi","year":"2022","journal-title":"Metrika"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"659","DOI":"10.3150\/08-BEJ168","article-title":"Local linear spatial quantile regression","volume":"15","author":"Hallin","year":"2009","journal-title":"Bernoulli"},{"key":"ref_44","first-page":"437","article-title":"Regression-Based Expected Shortfall Backtesting","volume":"20","author":"Bayer","year":"2022","journal-title":"J. Financ. Econom."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"590","DOI":"10.3390\/econometrics3030590","article-title":"A Kolmogorov-Smirnov based test for comparing the predictive accuracy of two sets of forecasts","volume":"3","author":"Hassani","year":"2015","journal-title":"Econometrics"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/10\/678\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:08:09Z","timestamp":1760112489000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/10\/678"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,9,30]]},"references-count":45,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2024,10]]}},"alternative-id":["axioms13100678"],"URL":"https:\/\/doi.org\/10.3390\/axioms13100678","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,9,30]]}}}