{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,10]],"date-time":"2026-10-10T04:15:01Z","timestamp":1791605701101,"version":"4.3.4"},"reference-count":21,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2008,10,14]],"date-time":"2008-10-14T00:00:00Z","timestamp":1223942400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2009,1,1]]},"abstract":"<jats:title>Summary<\/jats:title><jats:p>The paper proposes two Bayesian approaches to non-parametric monotone function estimation. The first approach uses a hierarchical Bayes framework and a characterization of smooth monotone functions given by Ramsay that allows unconstrained estimation. The second approach uses a Bayesian regression spline model of Smith and Kohn with a mixture distribution of constrained normal distributions as the prior for the regression coefficients to ensure the monotonicity of the resulting function estimate. The small sample properties of the two function estimators across a range of functions are provided via simulation and compared with existing methods. Asymptotic results are also given that show that Bayesian methods provide consistent function estimators for a large class of smooth functions. An example is provided involving economic demand functions that illustrates the application of the constrained regression spline estimator in the context of a multiple-regression model where two functions are constrained to be monotone.<\/jats:p>","DOI":"10.1111\/j.1467-9868.2008.00677.x","type":"journal-article","created":{"date-parts":[[2008,10,14]],"date-time":"2008-10-14T15:45:10Z","timestamp":1223999110000},"page":"159-175","source":"Crossref","is-referenced-by-count":76,"title":["A Bayesian Approach to Non-Parametric Monotone Function Estimation"],"prefix":"10.1093","volume":"71","author":[{"given":"Thomas S.","family":"Shively","sequence":"first","affiliation":[{"name":"University of Texas at Austin , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Thomas W.","family":"Sager","sequence":"additional","affiliation":[{"name":"University of Texas at Austin , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Stephen G.","family":"Walker","sequence":"additional","affiliation":[{"name":"University of Kent , Canterbury , UK"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2008,10,14]]},"reference":[{"key":"2023032921101402700_","volume-title":"Statistical Inference under Order Restrictions: the Theory and Application of Isotonic Regression","author":"Barlow","year":"1972"},{"key":"2023032921101402700_","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1080\/00031305.1992.10475878","article-title":"Explaining the Gibbs sampler","volume":"46","author":"Casella","year":"1992","journal-title":"Am. 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