{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,16]],"date-time":"2026-04-16T13:41:16Z","timestamp":1776346876222,"version":"3.51.2"},"reference-count":22,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2022,3,24]],"date-time":"2022-03-24T00:00:00Z","timestamp":1648080000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,3,24]],"date-time":"2022-03-24T00:00:00Z","timestamp":1648080000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100002341","name":"Academy of Finland","doi-asserted-by":"publisher","award":["298742"],"award-info":[{"award-number":["298742"]}],"id":[{"id":"10.13039\/501100002341","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100006136","name":"Teknologiateollisuuden 100-Vuotisjuhlas\u00e4\u00e4ti\u00f6","doi-asserted-by":"publisher","award":["70007503"],"award-info":[{"award-number":["70007503"]}],"id":[{"id":"10.13039\/501100006136","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Stat Comput"],"published-print":{"date-parts":[[2022,4,15]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Assessing goodness of fit to a given distribution plays an important role in computational statistics. The <jats:italic>probability integral transformation<\/jats:italic> (PIT) can be used to convert the question of whether a given sample originates from a reference distribution into a problem of testing for uniformity. We present new simulation- and optimization-based methods to obtain simultaneous confidence bands for the whole <jats:italic>empirical cumulative distribution function<\/jats:italic> (ECDF) of the PIT values under the assumption of uniformity. Simultaneous confidence bands correspond to such confidence intervals at each point that jointly satisfy a desired coverage. These methods can also be applied in cases where the reference distribution is represented only by a finite sample, which is useful, for example, for simulation-based calibration. The confidence bands provide an intuitive ECDF-based graphical test for uniformity, which also provides useful information on the quality of the discrepancy. We further extend the simulation and optimization methods to determine simultaneous confidence bands for testing whether multiple samples come from the same underlying distribution. This multiple sample comparison test is useful, for example, as a complementary diagnostic in multi-chain Markov chain Monte Carlo (MCMC) convergence diagnostics, where most currently used convergence diagnostics provide a single diagnostic value, but do not usually offer insight into the nature of the deviation. We provide numerical experiments to assess the properties of the tests using both simulated and real-world data and give recommendations on their practical application in computational statistics workflows.<\/jats:p>","DOI":"10.1007\/s11222-022-10090-6","type":"journal-article","created":{"date-parts":[[2022,3,25]],"date-time":"2022-03-25T10:03:03Z","timestamp":1648202583000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":38,"title":["Graphical test for discrete uniformity and its applications in goodness-of-fit evaluation and multiple sample comparison"],"prefix":"10.1007","volume":"32","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5249-348X","authenticated-orcid":false,"given":"Teemu","family":"S\u00e4ilynoja","sequence":"first","affiliation":[]},{"given":"Paul-Christian","family":"B\u00fcrkner","sequence":"additional","affiliation":[]},{"given":"Aki","family":"Vehtari","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,3,24]]},"reference":[{"issue":"4","key":"10090_CR1","doi-asserted-by":"publisher","first-page":"249","DOI":"10.1080\/00031305.2013.847865","volume":"67","author":"S Aldor-Noiman","year":"2013","unstructured":"Aldor-Noiman, S., Brown, L.D., Buja, A., Rolke, W., Stine, R.A.: The power to see: a new graphical test of normality. Am. Stat. 67(4), 249\u2013260 (2013). https:\/\/doi.org\/10.1080\/00031305.2013.847865","journal-title":"Am. Stat."},{"key":"10090_CR2","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719062","volume-title":"A First Course in Order Statistics (Classics in Applied Mathematics)","author":"BC Arnold","year":"2008","unstructured":"Arnold, B.C., Balakrishnan, N., Nagaraja, H.N.: A First Course in Order Statistics (Classics in Applied Mathematics). Society for Industrial and Applied Mathematics, USA (2008)"},{"key":"10090_CR3","first-page":"61","volume-title":"Algorithms for Minimization Without Derivatives","author":"R Brent","year":"1973","unstructured":"Brent, R.: An algorithm with guaranteed convergence for finding the minimum of a function of one variable. In: Algorithms for Minimization Without Derivatives, pp. 61\u201380. Prentice-Hall, Englewood Cliffs, NJ (1973)"},{"issue":"2","key":"10090_CR4","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1214\/ss\/1009213286","volume":"16","author":"LD Brown","year":"2001","unstructured":"Brown, L.D., Cai, T.T., DasGupta, A.: Interval estimation for a binomial proportion. Stat. Sci. 16(2), 101\u2013133 (2001). https:\/\/doi.org\/10.1214\/ss\/1009213286","journal-title":"Stat. Sci."},{"issue":"3","key":"10090_CR5","doi-asserted-by":"publisher","first-page":"675","DOI":"10.1198\/106186006X136976","volume":"15","author":"SR Cook","year":"2006","unstructured":"Cook, S.R., Gelman, A., Rubin, D.B.: Validation of software for Bayesian models using posterior quantiles. J. Comput. Graph. Stat. 15(3), 675\u2013692 (2006)","journal-title":"J. Comput. Graph. Stat."},{"issue":"4","key":"10090_CR6","doi-asserted-by":"publisher","first-page":"1254","DOI":"10.1111\/j.1541-0420.2009.01191.x","volume":"65","author":"C Czado","year":"2009","unstructured":"Czado, C., Gneiting, T., Held, L.: Predictive model assessment for count data. Biometrics 65(4), 1254\u20131261 (2009). https:\/\/doi.org\/10.1111\/j.1541-0420.2009.01191.x","journal-title":"Biometrics"},{"key":"10090_CR7","unstructured":"D\u2019Agostino, R.B., Stephens, M.A.: Goodness-of-Fit Techniques. Marcel Dekker Inc, USA (1986)"},{"issue":"4","key":"10090_CR8","doi-asserted-by":"publisher","first-page":"940","DOI":"10.1080\/10618600.2017.1377082","volume":"26","author":"A Gelman","year":"2017","unstructured":"Gelman, A.: Correction to Cook, Gelman, and Rubin (2006). J. Comput. Graph. Stat. 26(4), 940\u2013940 (2017). https:\/\/doi.org\/10.1080\/10618600.2017.1377082","journal-title":"J. Comput. Graph. Stat."},{"key":"10090_CR9","doi-asserted-by":"publisher","unstructured":"Gelman, A., Carlin, J.B., Stern, H.S., Dunson, D.B.: Aki VEHTARI a Donald B. RUBIN, 2013. Bayesian Data Anal. (2013). https:\/\/doi.org\/10.1201\/b16018","DOI":"10.1201\/b16018"},{"issue":"4","key":"10090_CR10","first-page":"457","volume":"7","author":"A Gelman","year":"1992","unstructured":"Gelman, A., Rubin, D.B.: Inference from iterative simulation using multiple sequences. Stat. Sci. 7(4), 457\u2013472 (1992)","journal-title":"Stat. Sci."},{"key":"10090_CR11","unstructured":"Gelman, A., Vehtari, A., Simpson, D., Margossian, C.C., Carpenter, B., Yao, Y., Kennedy, L., Gabry, J., B\u00fcrkner, P.C., Modr\u00e1k, M. (2020). Bayesian workflow. arXiv preprint arXiv:2011.01808"},{"issue":"2","key":"10090_CR12","doi-asserted-by":"publisher","first-page":"243","DOI":"10.1111\/j.1467-9868.2007.00587.x","volume":"69","author":"T Gneiting","year":"2007","unstructured":"Gneiting, T., Balabdaoui, F., Raftery, A.E.: Probabilistic forecasts, calibration and sharpness. J. Royal Stat. Soc. Series B (Stat. Methodol.) 69(2), 243\u2013268 (2007). https:\/\/doi.org\/10.1111\/j.1467-9868.2007.00587.x","journal-title":"J. Royal Stat. Soc. Series B (Stat. Methodol.)"},{"key":"10090_CR13","first-page":"83","volume":"4","author":"A Kolmogorov","year":"1933","unstructured":"Kolmogorov, A.: Sulla determinazione empirica di una lgge di distribuzione. Inst. Ital. Attuari, Giorn. 4, 83\u201391 (1933)","journal-title":"Inst. Ital. Attuari, Giorn."},{"key":"10090_CR14","doi-asserted-by":"publisher","unstructured":"Lambert, B., Vehtari, A.: $$R^{*}$$: A robust MCMC convergence diagnostic with uncertainty using decision tree classifiers. Bayesian Anal. https:\/\/doi.org\/10.1214\/20-BA1252","DOI":"10.1214\/20-BA1252"},{"issue":"427","key":"10090_CR15","doi-asserted-by":"publisher","first-page":"1000","DOI":"10.1080\/01621459.1994.10476834","volume":"89","author":"T Ledwina","year":"1994","unstructured":"Ledwina, T.: Data-driven version of Neyman\u2019s smooth test of fit. J. Am. Stat. Assoc. 89(427), 1000\u20131005 (1994). https:\/\/doi.org\/10.1080\/01621459.1994.10476834","journal-title":"J. Am. Stat. Assoc."},{"issue":"4","key":"10090_CR16","doi-asserted-by":"publisher","first-page":"315","DOI":"10.1080\/02331880500178562","volume":"39","author":"Y Marhuenda","year":"2005","unstructured":"Marhuenda, Y., Morales, D., Pardo, M.C.: A comparison of uniformity tests. Statistics 39(4), 315\u2013327 (2005). https:\/\/doi.org\/10.1080\/02331880500178562","journal-title":"Statistics"},{"issue":"253","key":"10090_CR17","doi-asserted-by":"publisher","first-page":"68","DOI":"10.1080\/01621459.1951.10500769","volume":"46","author":"FJ Massey Jr","year":"1951","unstructured":"Massey, F.J., Jr.: The Kolmogorov-Smirnov testfor goodness of fit. J. Am. Stat. Assoc. 46(253), 68\u201378 (1951)","journal-title":"J. Am. Stat. Assoc."},{"issue":"3\u20134","key":"10090_CR18","doi-asserted-by":"publisher","first-page":"149","DOI":"10.1080\/03461238.1937.10404821","volume":"1937","author":"J Neyman","year":"1937","unstructured":"Neyman, J.: Smooth test for goodness of fit. Scand. Actuar. J. 1937(3\u20134), 149\u2013199 (1937). https:\/\/doi.org\/10.1080\/03461238.1937.10404821","journal-title":"Scand. Actuar. J."},{"key":"10090_CR19","unstructured":"Stan Development Team. (n.d.). Stan user\u2019s guide. Retrieved September 29, 2020, from https:\/\/mc-stan.org\/docs\/2_24\/stan-users-guide\/index.html"},{"key":"10090_CR20","unstructured":"Talts, S., Betancourt, M., Simpson, D., Vehtari, A., & Gelman, A. (2020). Validating Bayesian inference algorithms with simulation-based calibration. arXiv preprint arXiv:1804.06788v2"},{"key":"10090_CR21","doi-asserted-by":"publisher","unstructured":"Vehtari, A., Gelman, A., Simpson, D., Carpenter, B., B\u00fcrkner, P.C.: Rank-normalization, folding, and localization: an improved $$\\widehat{R}$$ for assessing convergence of MCMC (with discussion). Bayesian Anal. 16(2), 667\u2013718 (2021). https:\/\/doi.org\/10.1214\/20-BA1221","DOI":"10.1214\/20-BA1221"},{"issue":"1\u20132","key":"10090_CR22","doi-asserted-by":"publisher","first-page":"109","DOI":"10.1093\/biomet\/48.1-2.109","volume":"48","author":"GS Watson","year":"1961","unstructured":"Watson, G.S.: Goodness-of-fit tests on a circle. Biometrika 48(1\u20132), 109\u2013114 (1961). https:\/\/doi.org\/10.1093\/biomet\/48.1-2.109","journal-title":"Biometrika"}],"container-title":["Statistics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-022-10090-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11222-022-10090-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-022-10090-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,4,21]],"date-time":"2022-04-21T14:40:43Z","timestamp":1650552043000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11222-022-10090-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,3,24]]},"references-count":22,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2022,4,15]]}},"alternative-id":["10090"],"URL":"https:\/\/doi.org\/10.1007\/s11222-022-10090-6","relation":{},"ISSN":["0960-3174","1573-1375"],"issn-type":[{"value":"0960-3174","type":"print"},{"value":"1573-1375","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,3,24]]},"assertion":[{"value":"19 March 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 February 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 March 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"32"}}