[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-37267-en":3,"doc-seo-37267-105":29,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},37267,1649267921044,"Ava Thompson","https://us-avatar.wpscdn.com/avatar/1800007509477c92dfb?_k=1782875107921204101",8,"Research & Report","Bayesian Lasso and Adaptive Lasso Expectile Regression","This article addresses regularization in expectile regression using a Bayesian framework. Two Bayesian regularization methods are proposed for covariate selection and estimation: Bayesian Lasso and adaptive Lasso expectile regression. Posterior inference is performed via two efficient Gibbs sampling algorithms based on a scale mixture of uniform (SMU) representation of the Laplace density. Simulation studies and two real data sets are used to illustrate performance, showing strong results across varied scenarios.","Communications in Statistics-Simulation and Computation  \nISSN: 0361-0918 (Print) 1532-4141 (Online) Journal [homepage: ](homepage: www.tandfonline.com/journals/lssp20)[www.tandfonline.com/journals/lssp20](homepage: www.tandfonline.com/journals/lssp20)  \nBayesian Lasso and adaptive Lasso expectile regression  \nRahim Alhamzawi  \nTo cite this article: Rahim Alhamzawi (26 Jun 2026): Bayesian Lasso and adaptive Lasso expectile regression, Communications in Statistics-Simulation and Computation, DOI: 10. 1080/03610918 .2026.2688382  \nTo link to this article: [https://doi.org/10.1080/03610918.2026.2688382](https://doi.org/10.1080/03610918.2026.2688382)  \n Published online: 26 Jun 2026.  \n\n|  Submit your article to this journal  |  |\n| --- | --- |\n|  | Article views: 9 |\n|  | View related articles  |\n|  View Crossmark data |  |\n\nFull Terms & Conditions of access and use can be found at [https://www.tandfonline.com/action/journalInformation?journalCode=lssp20](https://www.tandfonline.com/action/journalInformation?journalCode=lssp20)  \nCOMMUNICATIONS IN STATISTICS - SIMULATION AND COMPUTATIONVR  \n[https://doi.org/10.1080/03610918.2026.2688382](https://doi.org/10.1080/03610918.2026.2688382)  \nBayesian Lasso and adaptive Lasso expectile regression  \nRahim Alhamzawi   \nStatistics Department, College of Administration and Economics, Al-Qadisiyah University, Al Diwaniyah, Iraq  \nABSTRACT  \nThis article considers regularization in expectile regression from a Bayesian framework. Specifically, we proposed two Bayesian regularization approaches for covariate selection and estimation in expectile regression: the Bayesian Lasso and adaptive Lasso expectile regression. Two simple and efficient Gibbs sampling algorithms were developed for posterior inference using a scale mixture of uniform (SMU) representation of the Laplace density. The proposed approaches are illustrated via simulation studies and two real data sets. Compared to some of the existing approaches, results show that the proposed approaches perform very well under a variety of simulation studies and the real data sets.  \nARTICLE HISTORY  \nReceived 11 May 2025 Accepted 8 June 2026  \nKEYWORDS  \nAdaptive expectile regression; Gibbs sampler; Lasso; MCMC; Regularization Lasso  \n1. Introduction  \nExpectile regression (Newey and Powell 1987) has gained increasing popularity as it offers more information than standard mean regression. It can be seen as a generalization of the standard mean regression and an alternative method to quantile regression. There exists a large literature on expectile regression approaches, and we refer to Newey and Powell (1987), Seipp et al. (2021) and Waltrup et al. (2015) for an overview. Expectile regression models have been applied in many different areas: finance (Taylor 2007; Kuan et al. 2009), economics (Waltrup et al. 2015; Farooq and Steinwart 2017), survival analysis (Girard et al. 2021; Seipp et al. 2021), agricultural (Zhou et al. 2024), and so on.  \nLet yi ði, :::, nÞ denote the response variable, and let xi denote the corresponding covariate vector of the ith response. Then the linear expectile regression model for the sth expectile, s 2 ð0,1Þ, is yi ¼ x0ib þ ui: The expectile regression (ER) coefficient vector b is estimated by minimizing  \nn  \ni is1i1 wi ðyyyiii xxxbbb,:Þ2 , (1)  \nAs shown in Newey and Powell (1987), the ER estimates for the regression coefficients vector b can be achieved by weighted least squares (WLS) implementations. Extensions to more complex problem have been studied in recent literatures (De Rossi and Harvey 2009; Schnabel and Eilers 2009; Sobotka and Kneib 2012; Sobotka et al. 2013; Waltrup et al. 2015; Farooq and Steinwart 2017; Daouia et al. 2018; Bellini et al. 2021; Wang et al. 2025) . Although the large sample theory for expectile regression (ER) has been well studied (Zhang 1994; Girard et al.  \nCONTACT Alhamzawi Rahim  [ralhamzawi@yahoo.com](ralhamzawi@yahoo.com)  Statistics Department, College of Administration and ","cbCaivjjjb8hg0Io","https://ap.wps.com/l/cbCaivjjjb8hg0Io","pdf",2393306,1,21,"English","en",105,"# Introduction\n## Expectile regression background\n## Bayesian regularization approaches","[{\"question\":\"What problem does the article focus on in expectile regression?\",\"answer\":\"The article focuses on introducing regularization for expectile regression within a Bayesian framework to support covariate selection and estimation.\"},{\"question\":\"What Bayesian methods are proposed for expectile regression?\",\"answer\":\"The study proposes Bayesian Lasso expectile regression and adaptive Lasso expectile regression.\"},{\"question\":\"How is posterior inference carried out in the proposed methods?\",\"answer\":\"Two efficient Gibbs sampling algorithms are developed, using a scale mixture of uniform (SMU) representation of the Laplace density for posterior inference.\"}]",1783026128,53,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":27},"bayesian-lasso-and-adaptive-lasso-expectile-regression","",{"@graph":35,"@context":84},[36,53,67],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/bayesian-lasso-and-adaptive-lasso-expectile-regression/37267/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-02",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What problem does the article focus on in expectile regression?","Question",{"text":74,"@type":75},"The article focuses on introducing regularization for expectile regression within a Bayesian framework to support covariate selection and estimation.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"What Bayesian methods are proposed for expectile regression?",{"text":79,"@type":75},"The study proposes Bayesian Lasso expectile regression and adaptive Lasso expectile regression.",{"name":81,"@type":72,"acceptedAnswer":82},"How is posterior inference carried out in the proposed methods?",{"text":83,"@type":75},"Two efficient Gibbs sampling algorithms are developed, using a scale mixture of uniform (SMU) representation of the Laplace density for posterior inference.","https://schema.org",{"og:url":51,"og:type":86,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":88,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,104,109,114,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":101,"show_sort_weight":102,"slug":103},"Exam",70,"exam",{"id":105,"doc_module":4,"doc_module_name":45,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":45,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":45,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":45,"category_name":136,"show_sort_weight":105,"slug":137},19,"General","general"]