{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,6]],"date-time":"2026-10-06T00:59:26Z","timestamp":1791248366105,"version":"4.1.0"},"reference-count":29,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[2023,4,3]],"date-time":"2023-04-03T00:00:00Z","timestamp":1680480000000},"content-version":"vor","delay-in-days":6928,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS 99-71127"],"award-info":[{"award-number":["DMS 99-71127"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2004,5,1]]},"abstract":"<jats:title>Summary<\/jats:title><jats:p>Likelihood methods are often difficult to use with large, irregularly sited spatial data sets, owing to the computational burden. Even for Gaussian models, exact calculations of the likelihood for n observations require O(n3) operations. Since any joint density can be written as a product of conditional densities based on some ordering of the observations, one way to lessen the computations is to condition on only some of the \u2018past\u2019 observations when computing the conditional densities. We show how this approach can be adapted to approximate the restricted likelihood and we demonstrate how an estimating equations approach allows us to judge the efficacy of the resulting approximation. Previous work has suggested conditioning on those past observations that are closest to the observation whose conditional density we are approximating. Through theoretical, numerical and practical examples, we show that there can often be considerable benefit in conditioning on some distant observations as well.<\/jats:p>","DOI":"10.1046\/j.1369-7412.2003.05512.x","type":"journal-article","created":{"date-parts":[[2004,12,28]],"date-time":"2004-12-28T19:41:00Z","timestamp":1104262860000},"page":"275-296","source":"Crossref","is-referenced-by-count":301,"title":["Approximating Likelihoods for Large Spatial Data Sets"],"prefix":"10.1093","volume":"66","author":[{"given":"Michael L.","family":"Stein","sequence":"first","affiliation":[{"name":"University of Chicago , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhiyi","family":"Chi","sequence":"additional","affiliation":[{"name":"University of Chicago , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Leah J.","family":"Welty","sequence":"additional","affiliation":[{"name":"University of Chicago , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2004,4,14]]},"reference":[{"key":"2023040400410385100_","volume-title":"Handbook of Mathematical Functions","author":"Abramowitz","year":"1965","edition":"9th"},{"key":"2023040400410385100_","doi-asserted-by":"crossref","DOI":"10.1002\/9780470316993","volume-title":"Geostatistics: Modeling Spatial Uncertainty","author":"Chil\u00e8s","year":"1999"},{"key":"2023040400410385100_","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-2477-6","volume-title":"Plane Answers to Complex Questions: the Theory of Linear Models","author":"Christensen","year":"1996","edition":"2nd"},{"key":"2023040400410385100_","first-page":"299","article-title":"Model-based geostatistics (with discussion)","volume":"47","author":"Diggle","year":"1998","journal-title":"Appl. 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