{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T12:28:57Z","timestamp":1787228937170,"version":"build-2736575974"},"reference-count":20,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM Rev."],"published-print":{"date-parts":[[2015,1]]},"abstract":"<jats:p>Most mathematical models include parameters that need to be determined from measurements. The estimated values of these parameters and their uncertainties depend on assumptions made about noise levels, models, or prior knowledge. But what can we say about the validity of such estimates, and the influence of these assumptions? This paper is concerned with methods to address these questions, and for didactic purposes it is written in the context of a concrete nonlinear parameter estimation problem. We will use the results of a physical experiment conducted by Allmaras et al. at Texas A&amp;M University [M. Allmaras et al., SIAM Rev., 55 (2013), pp. 149--167] to illustrate the importance of validation procedures for statistical parameter estimation. We describe statistical methods and data analysis tools to check the choices of likelihood and prior distributions, and provide examples of how to compare Bayesian results with those obtained by non-Bayesian methods based on different types of assumptions. We explain how different statistical methods can be used in complementary ways to improve the understanding of parameter estimates and their uncertainties.<\/jats:p>","DOI":"10.1137\/130929230","type":"journal-article","created":{"date-parts":[[2015,2,5]],"date-time":"2015-02-05T21:04:40Z","timestamp":1423170280000},"page":"131-149","source":"Crossref","is-referenced-by-count":18,"title":["Statistics of Parameter Estimates: A Concrete Example"],"prefix":"10.1137","volume":"57","author":[{"given":"Oscar","family":"Aguilar","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Moritz","family":"Allmaras","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wolfgang","family":"Bangerth","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Luis","family":"Tenorio","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2015,2,5]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/100788604"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1214\/ss\/1009213286"},{"key":"atypb3","doi-asserted-by":"crossref","unstructured":"B. P. Carlin and T. A. Louis,\n                      Bayesian Methods for Data Analysis\n                      , 3rd ed., CRC Press, Boca Raton, FL, 2008.","DOI":"10.1201\/b14884"},{"key":"atypb4","unstructured":"G. Casella and R. L. Berger,\n                      Statistical Inference\n                      , 2nd ed., Duxbury, Pacific Grove, CA, 2002."},{"key":"atypb5","doi-asserted-by":"crossref","unstructured":"A. C. Davison and D. V. Hinkley,\n                      Bootstrap Methods and Their Applications\n                      , Cambridge University Press, Cambridge, UK, 1997.","DOI":"10.1017\/CBO9780511802843"},{"key":"atypb6","doi-asserted-by":"crossref","unstructured":"N. R. Draper and H. Smith,\n                      Applied Regression Analysis\n                      , 3rd ed., Wiley, New York, 1998.","DOI":"10.1002\/9781118625590"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1093\/biomet\/65.3.457"},{"key":"atypb8","doi-asserted-by":"crossref","unstructured":"B. Efron and R. Tibshirani,\n                      An Introduction to the Bootstrap\n                      , Chapman & Hall, London, 1993.","DOI":"10.1007\/978-1-4899-4541-9"},{"key":"atypb9","doi-asserted-by":"crossref","unstructured":"T. S. Ferguson,\n                      A Course in Large Sample Theory\n                      , Chapman & Hall, London, 1996.","DOI":"10.1007\/978-1-4899-4549-5"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1007\/BF00208723"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1017938917"},{"key":"atypb13","doi-asserted-by":"crossref","unstructured":"A. Gelman, J. Carlin, H. S. Stern, and D. B. Rubin,\n                      Bayesian Data Analysis\n                      , 2nd ed., Chapman & Hall, London, 2003.","DOI":"10.1201\/9780429258480"},{"key":"atypb14","unstructured":"R. A. Johnson and D. W. Wichern,\n                      Applied Multivariate Analysis\n                      , 6th ed., Prentice-Hall, Englewood Cliffs, NJ, 2007."},{"key":"atypb15","unstructured":"E. L. Lehmann and J. P. Romano,\n                      Testing Statistical Hypotheses\n                      , Springer, New York, 2005."},{"key":"atypb16","unstructured":"J. Rice,\n                      Mathematical Statistics and Data Analysis\n                      , 3rd ed., Duxbury, Belmont, CA, 2007."},{"key":"atypb17","doi-asserted-by":"crossref","unstructured":"F. J. Samaniego,\n                      A Comparison of the Bayesian and Frequentist Approaches to Estimation\n                      , Springer, New York, 2010.","DOI":"10.1007\/978-1-4419-5941-6"},{"key":"atypb18","doi-asserted-by":"crossref","unstructured":"G. A. F. Seber and C. J. Wild,\n                      Nonlinear Regression\n                      , Wiley, New York, 2003.","DOI":"10.1002\/9780471722199"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1029\/92JB00739"},{"key":"atypb20","first-page":"9","author":"Stark P. B.","year":"2011","journal-title":"Wiley"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/27\/4\/045001"}],"container-title":["SIAM Review"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/130929230","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T12:02:20Z","timestamp":1787227340000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/130929230"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,1]]},"references-count":20,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2015,1]]}},"alternative-id":["10.1137\/130929230"],"URL":"https:\/\/doi.org\/10.1137\/130929230","relation":{},"ISSN":["0036-1445","1095-7200"],"issn-type":[{"value":"0036-1445","type":"print"},{"value":"1095-7200","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,1]]}}}