{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,7]],"date-time":"2026-08-07T16:49:49Z","timestamp":1786121389589,"version":"build-2736575974"},"reference-count":46,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2026,8,5]],"date-time":"2026-08-05T00:00:00Z","timestamp":1785888000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Key Research and Development Program of China","award":["2021YFA1000102"],"award-info":[{"award-number":["2021YFA1000102"]}]},{"DOI":"10.13039\/501100007129","name":"Natural Science Foundation of Shandong Province","doi-asserted-by":"crossref","award":["ZR2024MA074"],"award-info":[{"award-number":["ZR2024MA074"]}],"id":[{"id":"10.13039\/501100007129","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100013139","name":"Humanities and Social Science Fund of Ministry of Education of China","doi-asserted-by":"crossref","award":["24YJA910003"],"award-info":[{"award-number":["24YJA910003"]}],"id":[{"id":"10.13039\/501100013139","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This paper develops and evaluates HQER-SFA, a likelihood-based stochastic frontier specification that embeds a hybrid quantile-expectile error layer into the bilateral noise component while preserving the standard one-sided inefficiency structure. The model nests Quantile-SFA when \u03b3=0 and extends it by normalizing the hybrid loss into a proper bilateral error density, so that likelihood inference, residual decomposition, and technical-efficiency recovery remain in a unified stochastic-frontier framework. We compare HQER-SFA with Traditional-SFA and Quantile-SFA using three processed production modules, Monte Carlo parameter-inversion experiments, an expanded R=300 robustness design, \u03b3 sensitivity and ablation checks, convergence diagnostics, and a source-assisted small-sample extension. The results demonstrate clear gains for technical-efficiency recovery: in the R=300 design, HQER-SFA attains win rates of 0.5646 for technical-efficiency RMSE and 0.5578 for technical-efficiency rank correlation, and the agricultural module shows the strongest real-data improvement under the flexible bilateral error layer. The source-assisted analysis further shows that same-domain initialization improves small-sample validation RMSE by about 2.43%, while excessive source-centered penalties should be controlled. Overall, HQER-SFA provides an interpretable and computationally feasible extension for technical-efficiency recovery, especially when bilateral noise is asymmetric, tail-sensitive, or heterogeneous across production modules.<\/jats:p>","DOI":"10.3390\/e28080881","type":"journal-article","created":{"date-parts":[[2026,8,5]],"date-time":"2026-08-05T10:41:42Z","timestamp":1785926502000},"page":"881","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Hybrid Quantile-Expectile Error Layers for Technical-Efficiency Recovery in Stochastic Frontier Analysis"],"prefix":"10.3390","volume":"28","author":[{"given":"Shengming","family":"Wang","sequence":"first","affiliation":[{"name":"College of Science, China University of Petroleum, Qingdao 266580, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yunquan","family":"Song","sequence":"additional","affiliation":[{"name":"College of Science, China University of Petroleum, Qingdao 266580, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Juan","family":"Yu","sequence":"additional","affiliation":[{"name":"College of Science, China University of Petroleum, Qingdao 266580, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,8,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/0304-4076(77)90052-5","article-title":"Formulation and Estimation of Stochastic Frontier Production Function Models","volume":"6","author":"Aigner","year":"1977","journal-title":"J. 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