{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T16:41:50Z","timestamp":1777653710286,"version":"3.51.4"},"reference-count":37,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2018,6,14]],"date-time":"2018-06-14T00:00:00Z","timestamp":1528934400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>One natural way to measure model adequacy is by using statistical distances as loss functions. A related fundamental question is how to construct loss functions that are scientifically and statistically meaningful. In this paper, we investigate non-quadratic distances and their role in assessing the adequacy of a model and\/or ability to perform model selection. We first present the definition of a statistical distance and its associated properties. Three popular distances, total variation, the mixture index of fit and the Kullback-Leibler distance, are studied in detail, with the aim of understanding their properties and potential interpretations that can offer insight into their performance as measures of model misspecification. A small simulation study exemplifies the performance of these measures and their application to different scientific fields is briefly discussed.<\/jats:p>","DOI":"10.3390\/e20060464","type":"journal-article","created":{"date-parts":[[2018,6,14]],"date-time":"2018-06-14T11:06:06Z","timestamp":1528974366000},"page":"464","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Non-Quadratic Distances in Model Assessment"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1453-8229","authenticated-orcid":false,"given":"Marianthi","family":"Markatou","sequence":"first","affiliation":[{"name":"Department of Biostatistics, University at Buffalo, Buffalo, NY 14214, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1928-2381","authenticated-orcid":false,"given":"Yang","family":"Chen","sequence":"additional","affiliation":[{"name":"Department of Biostatistics, University at Buffalo, Buffalo, NY 14214, USA"}]}],"member":"1968","published-online":{"date-parts":[[2018,6,14]]},"reference":[{"key":"ref_1","unstructured":"Taper, M.L., and Lele, S.R. (2004). Statistical distances as loss functions in assessing model adequacy. The Nature of Scientific Evidence: Statistical, Philosophical and Empirical Considerations, The University of Chicago Press."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"983","DOI":"10.1214\/009053607000000956","article-title":"Quadratic distances on probabilities: A unified foundation","volume":"36","author":"Lindsay","year":"2008","journal-title":"Ann. Stat."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Chen, D.G., Jin, Z., Li, G., Li, Y., Liu, A., and Zhao, Y. (2017). Statistical distances and their role in robustness. New Advances in Statistics and Data Science, Springer.","DOI":"10.1007\/978-3-319-69416-0"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"248","DOI":"10.1177\/0049124196025002005","article-title":"A note on calculating the \u03c0* index of fit for the analysis of contingency tables","volume":"25","author":"Xi","year":"1996","journal-title":"Sociol. Methods Res."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"587","DOI":"10.1214\/aos\/1176350821","article-title":"Pathologies of some minimum distance estimators","volume":"16","author":"Donoho","year":"1988","journal-title":"Ann. Stat."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1007\/BF02949773","article-title":"On the theory of statistical decision functions","volume":"3","author":"Matusita","year":"1951","journal-title":"Ann. Inst. Stat. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"631","DOI":"10.1214\/aoms\/1177728422","article-title":"Decision rules, based on the distance, for problems of fit, two samples, and estimation","volume":"26","author":"Matusita","year":"1955","journal-title":"Ann. Math. Stat."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"623","DOI":"10.1111\/j.2517-6161.1994.tb02004.x","article-title":"A new index of fit based on mixture methods for the analysis of contingency tables","volume":"56","author":"Rudas","year":"1994","journal-title":"J. Royal Stat. Soc. Series B"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"3644","DOI":"10.1214\/08-AOS603","article-title":"Building and using semiparametric tolerance regions for parametric multinomial models","volume":"37","author":"Liu","year":"2009","journal-title":"Ann. Stat."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1348\/000711003321645304","article-title":"Applications and computational strategies for the two-point mixture index of fit","volume":"56","author":"Dayton","year":"2003","journal-title":"Br. J. Math. Stat. Psychol."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"432","DOI":"10.1007\/s10958-014-1925-9","article-title":"On the robustness of mixture index of fit","volume":"200","author":"Verdes","year":"2014","journal-title":"J. Math. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1214\/aoms\/1177729694","article-title":"On information and sufficiency","volume":"22","author":"Kullback","year":"1951","journal-title":"Ann. Math. Stat."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"716","DOI":"10.1109\/TAC.1974.1100705","article-title":"A new look at the statistical model identification","volume":"19","author":"Akaike","year":"1974","journal-title":"IEEE Trans. Automat. Contr."},{"key":"ref_14","unstructured":"Eu\u00e1n, C., Ortega, J., and Esteban, P.C.A. (July, January 30). Detecting Changes in Wave Spectra Using the Total Variation Distance. Proceedings of the 23rd International Offshore and Polar Engineering Conference. International Society of Offshore and Polar Engineers, Anchorage, AK, USA."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1002\/env.2398","article-title":"Time series clustering using the total variation distance with applications in oceanography","volume":"27","author":"Ortega","year":"2016","journal-title":"Environmetrics"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"31","DOI":"10.2307\/1165237","article-title":"Estimating the importance of differential item functioning","volume":"22","author":"Rudas","year":"1997","journal-title":"J. Educ. Behav. Stat."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Dayton, M.C. (1999). Latent Class Scaling Analysis, Sage.","DOI":"10.4135\/9781412984720"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1348\/000711005X72818","article-title":"Testing the Rasch model by means of the mixture fit index","volume":"59","author":"Formann","year":"2006","journal-title":"Br. J. Math. Stat. Psychol."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1348\/000711006X136843","article-title":"Estimating the \u03c0* goodness of fit index for finite mixtures of item response models","volume":"61","author":"Revuelta","year":"2008","journal-title":"Br. J. Math. Stat. Psychol."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1177\/014662169001400305","article-title":"Rasch models in latent classes: An integration of two approaches to item analysis","volume":"14","author":"Rost","year":"1990","journal-title":"Appl. Psychol. Meas."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1111\/j.2044-8317.1991.tb00951.x","article-title":"A logistic mixture distribution model for polychotomous item responses","volume":"44","author":"Rost","year":"1991","journal-title":"Br. J. Math. Stat. Psychol."},{"key":"ref_22","unstructured":"Burnham, K.P., and Anderson, D.R. (2002). Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach, Springer."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"553","DOI":"10.1016\/j.jeconom.2005.07.005","article-title":"Markov-switching model selection using Kullback\u2013Leibler divergence","volume":"134","author":"Smith","year":"2006","journal-title":"J. Econom."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1093\/ije\/28.3.521","article-title":"Selecting diagnostic tests for ruling out or ruling in disease: the use of the Kullback-Leibler distance","volume":"28","author":"Lee","year":"1999","journal-title":"Int. J. Epidemiol."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1500","DOI":"10.1016\/S0140-6736(05)66422-7","article-title":"Refining clinical diagnosis with likelihood ratios","volume":"365","author":"Grimes","year":"2005","journal-title":"Lancet"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Cliff, O.M., Prokopenko, M., and Fitch, R. (2018). Minimising the Kullback\u2013Leibler Divergence for Model Selection in Distributed Nonlinear Systems. Entropy, 20.","DOI":"10.3390\/e20020051"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"314","DOI":"10.22237\/jmasm\/1272688260","article-title":"JMASM30 PI-LCA: A SAS program computing the two-point mixture index of fit for two-class LCA Models with dichotomous variables (SAS)","volume":"9","author":"Zhang","year":"2010","journal-title":"J. Mod. Appl. Stat. Methods"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"314","DOI":"10.22237\/jmasm\/1304223300","article-title":"Factors influencing the mixture index of model fit in contingency tables showing indenpendence","volume":"10","author":"Pan","year":"2011","journal-title":"J. Mod. Appl. Stat. Methods"},{"key":"ref_29","first-page":"471","article-title":"Finding and characterization of local optima in the \u03c0* problem for two-way contingency tables","volume":"36","author":"Verdes","year":"2000","journal-title":"Stud. Sci. Math. Hung."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"200","DOI":"10.1016\/0041-5553(67)90040-7","article-title":"The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming","volume":"7","author":"Bregman","year":"1967","journal-title":"USSR Comput. Math. Math. Phys."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"440","DOI":"10.1111\/j.2517-6161.1984.tb01318.x","article-title":"Multinomial goodness-of-fit tests","volume":"46","author":"Cressie","year":"1984","journal-title":"J. Royal Stat. Soc. Series B"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1093\/biomet\/85.3.549","article-title":"Robust and efficient estimation by minimising a density power divergence","volume":"85","author":"Basu","year":"1998","journal-title":"Biometrika"},{"key":"ref_33","unstructured":"Pardo, L. (2006). Statistical Inference Based on Divergence Measures, Chapman and Hall\/CRC."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Basu, A., Shioya, H., and Park, C. (2011). Statistical Inference: The Minimum Distance Approach, Chapman and Hall\/CRC.","DOI":"10.1201\/b10956"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Reiczigel, J., Isp\u00e1ny, M., Tusn\u00e1dy, G., Michaletzky, G., and Marozzi, M. (2017). Bias-corrected estimation of the Rudas\u2013Clogg\u2013Lindsay mixture index of fit. Br. J. Math. Stat. Psychol.","DOI":"10.1111\/bmsp.12118"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Ghosh, A., and Basu, A. (2018). A new family of divergences originating from model adequacy tests and application to robust statistical inference. IEEE Trans. Inf. Theory.","DOI":"10.1109\/TIT.2018.2794537"},{"key":"ref_37","unstructured":"Dimova, R., Markatou, M., and Afendras, G. (2018). Model Selection Based on the Relative Quadratic Risk, Department of Biostatistics, University at Buffalo. Technical Report."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/6\/464\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:08:46Z","timestamp":1760195326000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/6\/464"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,6,14]]},"references-count":37,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2018,6]]}},"alternative-id":["e20060464"],"URL":"https:\/\/doi.org\/10.3390\/e20060464","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,6,14]]}}}