{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:18:34Z","timestamp":1760149114374,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2023,7,4]],"date-time":"2023-07-04T00:00:00Z","timestamp":1688428800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Council for Scientific and Technological Development (CNPq)","award":["303192\/2022-4","1200525"],"award-info":[{"award-number":["303192\/2022-4","1200525"]}]},{"name":"Comiss\u00e3o de Aperfei\u00e7oamento de Pessoal do N\u00edvel Superior (CAPES)","award":["303192\/2022-4","1200525"],"award-info":[{"award-number":["303192\/2022-4","1200525"]}]},{"name":"FONDECYT","award":["303192\/2022-4","1200525"],"award-info":[{"award-number":["303192\/2022-4","1200525"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The data envelopment analysis is related to a non-parametric mathematical tool used to assess the relative efficiency of productive units. In different studies on productive efficiency, it is common to employ semi-parametric procedures in two stages to determine whether any exogenous factors of interest affect the performance of productive units. However, some of these procedures, particularly those based on conventional statistical inference, generate inconsistent estimates when dealing with incoherent data-generating processes. This inconsistency arises due to the efficiency scores being limited to the unit interval, and the estimated scores often exhibit serial correlation and have limited observations. To address such inconsistency, several strategies have been suggested, with the most well-known being an algorithm based on a parametric bootstrap procedure using the truncated normal distribution and its regression model. In this work, we present a modification of this algorithm that utilizes the beta distribution and its regression structure. The beta model allows for better accommodation of asymmetry in the data distribution. Our proposed algorithm introduces inferential characteristics that are superior to the original algorithm, resulting in a more statistically coherent data-generating process and improving the consistency property. We have conducted computational experiments that demonstrate the improved results achieved by our proposal.<\/jats:p>","DOI":"10.3390\/sym15071362","type":"journal-article","created":{"date-parts":[[2023,7,5]],"date-time":"2023-07-05T00:33:42Z","timestamp":1688517222000},"page":"1362","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Robust Semi-Parametric Inference for Two-Stage Production Models: A Beta Regression Approach"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9884-9090","authenticated-orcid":false,"given":"Raydonal","family":"Ospina","sequence":"first","affiliation":[{"name":"Department of Statistics, CASTLab, Universidade Federal de Pernambuco, Recife 50670-901, Brazil"},{"name":"Department of Statistics, IME, Universidade Federal da Bahia, Salvador 40170-110, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8780-3345","authenticated-orcid":false,"given":"Samuel G. F.","family":"Baltazar","sequence":"additional","affiliation":[{"name":"Department of Statistics, CASTLab, Universidade Federal de Pernambuco, Recife 50670-901, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4755-3270","authenticated-orcid":false,"given":"V\u00edctor","family":"Leiva","sequence":"additional","affiliation":[{"name":"School of Industrial Engineering, Pontificia Universidad Cat\u00f3lica de Valpara\u00edso, Valpara\u00edso 2362807, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2152-3199","authenticated-orcid":false,"given":"Jorge","family":"Figueroa-Z\u00fa\u00f1iga","sequence":"additional","affiliation":[{"name":"Department of Statistics, Universidad de Concepci\u00f3n, Concepci\u00f3n 4070386, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9897-8186","authenticated-orcid":false,"given":"Cecilia","family":"Castro","sequence":"additional","affiliation":[{"name":"Centre of Mathematics, University of Minho, 4710-057 Braga, Portugal"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,7,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"429","DOI":"10.1016\/0377-2217(78)90138-8","article-title":"Measuring the efficiency of decision making units","volume":"2","author":"Charnes","year":"1978","journal-title":"Eur. J. Oper. Res."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Wong, W.P. (2021). A global search method for inputs and outputs in data envelopment analysis: Procedures and managerial perspectives. Symmetry, 13.","DOI":"10.3390\/sym13071155"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1078","DOI":"10.1287\/mnsc.30.9.1078","article-title":"Some models for estimating technical and scale inefficiencies in data envelopment analysis","volume":"30","author":"Banker","year":"1984","journal-title":"Manag. Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/j.jeconom.2005.07.009","article-title":"Estimation and inference in two-stage, semi-parametric models of production processes","volume":"136","author":"Simar","year":"2007","journal-title":"J. Econom."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"893","DOI":"10.1016\/j.omega.2012.11.004","article-title":"A survey of DEA applications","volume":"41","author":"Liu","year":"2013","journal-title":"Omega"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"L\u00f3pez-Penabad, M.-C., Maside-Sanfiz, J.M., Torrelles, M.J., and Iglesias-Casal, A. (2020). Application of the DEA double bootstrap to analyze efficiency in Galician sheltered workshops. Sustainability, 12.","DOI":"10.3390\/su12166625"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3411","DOI":"10.1007\/s12351-021-00667-w","article-title":"Estimating the confidence intervals for DEA efficiency scores of Asia-Pacific airlines","volume":"22","author":"Ngo","year":"2022","journal-title":"Oper. Res. Int. J."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1016\/S0378-4266(00)00171-0","article-title":"Trends in relationship lending and factors affecting relationship lending efficiency","volume":"26","author":"Stanton","year":"2002","journal-title":"J. Bank. Financ."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2337","DOI":"10.1016\/j.eswa.2013.09.031","article-title":"Two-stage DEA: An application to major Brazilian banks","volume":"41","author":"Wanke","year":"2014","journal-title":"Expert Syst. Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"910","DOI":"10.1016\/j.mcm.2005.12.011","article-title":"A two-stage DEA model to evaluate the overall performance of Canadian life and health insurance companies","volume":"43","author":"Yang","year":"2006","journal-title":"Math. Comput. Model."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1037","DOI":"10.1287\/mnsc.34.9.1037","article-title":"The effect of unions on productivity: US surface mining of coal","volume":"34","author":"Byrnes","year":"1988","journal-title":"Manag. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"24","DOI":"10.2307\/1907382","article-title":"Estimation of relationships for limited dependent variables","volume":"26","author":"Tobin","year":"1958","journal-title":"Econometrica"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1016\/j.ejor.2006.05.019","article-title":"Second stage DEA: Comparison of approaches for modelling the DEA score","volume":"181","author":"Hoff","year":"2007","journal-title":"Eur. J. Oper. Res."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1191\/1471082X03st053oa","article-title":"Regression analysis of variates observed on (0, 1): Percentages, proportions and fractions","volume":"3","author":"Kieschnick","year":"2003","journal-title":"Stat. Model."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1214\/aos\/1176344552","article-title":"Bootstrap methods: Another look at the Jackknife","volume":"7","author":"Efron","year":"1979","journal-title":"Ann. Stat."},{"key":"ref_16","unstructured":"Hirschberg, J.G., and Lloyd, P.J. (2000). An Application of Post-DEA Bootstrap Regression Analysis to the Spill over of the Technology of Foreign-Invested Enterprises in China, Department of Economics, University of Melbourne. Technical Report."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Fried, H.O., Lovell, C.A.K., and Schmidt, S.S. (2008). The Measurement of Productive Efficiency and Productivity Growth, Oxford University Press.","DOI":"10.1093\/acprof:oso\/9780195183528.001.0001"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"985","DOI":"10.1057\/palgrave.jors.2601400","article-title":"Including principal component weights to improve discrimination in data envelopment analysis","volume":"53","author":"Adler","year":"2002","journal-title":"J. Oper. Res. Soc."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"799","DOI":"10.1080\/0266476042000214501","article-title":"Beta regression for modelling rates and proportions","volume":"31","author":"Ferrari","year":"2004","journal-title":"J. Appl. Stat."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Couri, L., Ospina, R., da Silva, G., Leiva, V., and Figueroa-Z\u00fa\u00f1iga, J. (2022). A study on computational algorithms in the estimation of parameters for a class of beta regression models. Mathematics, 10.","DOI":"10.3390\/math10030299"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1002\/asmb.2278","article-title":"A beta partial least squares regression model: Diagnostics and application to mining industry data","volume":"34","author":"Huerta","year":"2018","journal-title":"Appl. Stoch. Model. Bus. Ind."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Davison, A.C., and Hinkley, D.V. (1997). Bootstrap Methods and Their Application, Cambridge University Press.","DOI":"10.1017\/CBO9780511802843"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"7161628","DOI":"10.1155\/2020\/7161628","article-title":"An interval efficiency measurement in DEA when considering undesirable outputs","volume":"2020","author":"Mo","year":"2020","journal-title":"Complexity"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1093\/oxfordjournals.pan.a004873","article-title":"Maximum likelihood estimation of models with beta-distributed dependent variables","volume":"9","author":"Paolino","year":"2001","journal-title":"Political Anal."},{"key":"ref_25","first-page":"13","article-title":"Improved maximum likelihood estimation in a new class of beta regression models","volume":"19","author":"Vasconcellos","year":"2005","journal-title":"Braz. J. Probab. Stat."},{"key":"ref_26","first-page":"387","article-title":"Modeling heavy-tailed bounded data by the trapezoidal beta distribution with applications","volume":"20","author":"Leiva","year":"2022","journal-title":"Revstat-Stat. J."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"919","DOI":"10.1007\/s00362-021-01260-1","article-title":"Robust beta regression modeling with errors-in-variables: A Bayesian approach and numerical applications","volume":"63","author":"Bayes","year":"2022","journal-title":"Stat. Pap."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Altun, E., and El-Morshedy, M. (2021). SimBetaReg web-tool: The easiest way to implement the beta and simplex regression models. Symmetry, 13.","DOI":"10.3390\/sym13122437"},{"key":"ref_29","first-page":"153","article-title":"The likelihood ratio, Wald, and Lagrange multiplier tests: An expository note","volume":"36","author":"Buse","year":"1982","journal-title":"Am. Stat."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"370","DOI":"10.2307\/2344614","article-title":"Generalized linear models","volume":"135","author":"Nelder","year":"1972","journal-title":"J. R. Stat. Soc. A"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"348","DOI":"10.1016\/j.csda.2009.08.017","article-title":"Improved estimators for a general class of beta regression models","volume":"54","author":"Simas","year":"2010","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1827","DOI":"10.1080\/00949655.2011.599033","article-title":"Testing inference in variable dispersion beta regressions","volume":"82","author":"Souza","year":"2012","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_33","first-page":"817","article-title":"A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity","volume":"48","author":"White","year":"1980","journal-title":"Econom. J. Econom. Soc."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18637\/jss.v011.i10","article-title":"Econometric computing with HC and HAC covariance matrix estimators","volume":"11","author":"Zeileis","year":"2004","journal-title":"J. Stat. Softw."},{"key":"ref_35","unstructured":"Naumann, F., Freytag, J., and Spiliopoulou, M. (1998). Proceedings of the 3rd Conference on Information Quality, MIT Sloan School of Management."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"336","DOI":"10.1214\/aos\/1176349860","article-title":"Finite sample properties and asymptotic efficiency of Monte Carlo tests","volume":"14","author":"Jockel","year":"1986","journal-title":"Ann. Stat."},{"key":"ref_37","unstructured":"Lehmann, E.L., and Romano, J.P. (2005). Testing Statistical Hypotheses, Springer."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Figueroa-Z\u00fa\u00f1iga, J., Toledo, J.G., Lagos-Alvarez, B., Leiva, V., and Navarrete, J.P. (2023). Inference based on the stochastic expectation-maximization algorithm in a Kumaraswamy model with an application to COVID-19 cases in Chile. Mathematics, 11.","DOI":"10.3390\/math11132894"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1609","DOI":"10.1016\/j.csda.2011.10.005","article-title":"A general class of zero-or-one inflated beta regression models","volume":"56","author":"Ospina","year":"2012","journal-title":"Comput. Stat. Data Anal."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/7\/1362\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:06:00Z","timestamp":1760126760000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/7\/1362"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,4]]},"references-count":39,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2023,7]]}},"alternative-id":["sym15071362"],"URL":"https:\/\/doi.org\/10.3390\/sym15071362","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,7,4]]}}}