{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:16:40Z","timestamp":1760242600901,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2017,11,20]],"date-time":"2017-11-20T00:00:00Z","timestamp":1511136000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS\u20131712418"],"award-info":[{"award-number":["DMS\u20131712418"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["grants 11690014"],"award-info":[{"award-number":["grants 11690014"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The classical quadratic loss for the partially linear model (PLM) and the likelihood function for the generalized PLM are not resistant to outliers. This inspires us to propose a class of \u201crobust-Bregman divergence (BD)\u201d estimators of both the parametric and nonparametric components in the general partially linear model (GPLM), which allows the distribution of the response variable to be partially specified, without being fully known. Using the local-polynomial function estimation method, we propose a computationally-efficient procedure for obtaining \u201crobust-BD\u201d estimators and establish the consistency and asymptotic normality of the \u201crobust-BD\u201d estimator of the parametric component \r\n          \r\n            \r\n              \r\n                \u03b2\r\n                o\r\n              \r\n            \r\n          \r\n        . For inference procedures of \r\n          \r\n            \r\n              \r\n                \u03b2\r\n                o\r\n              \r\n            \r\n          \r\n         in the GPLM, we show that the Wald-type test statistic \r\n          \r\n            \r\n              \r\n                W\r\n                n\r\n              \r\n            \r\n          \r\n         constructed from the \u201crobust-BD\u201d estimators is asymptotically distribution free under the null, whereas the likelihood ratio-type test statistic \r\n          \r\n            \r\n              \r\n                \u039b\r\n                n\r\n              \r\n            \r\n          \r\n         is not. This provides an insight into the distinction from the asymptotic equivalence (Fan and Huang 2005) between \r\n          \r\n            \r\n              \r\n                W\r\n                n\r\n              \r\n            \r\n          \r\n         and \r\n          \r\n            \r\n              \r\n                \u039b\r\n                n\r\n              \r\n            \r\n          \r\n         in the PLM constructed from profile least-squares estimators using the non-robust quadratic loss. Numerical examples illustrate the computational effectiveness of the proposed \u201crobust-BD\u201d estimators and robust Wald-type test in the appearance of outlying observations.<\/jats:p>","DOI":"10.3390\/e19110625","type":"journal-article","created":{"date-parts":[[2017,11,20]],"date-time":"2017-11-20T11:35:45Z","timestamp":1511177745000},"page":"625","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Robust-BD Estimation and Inference for General Partially Linear Models"],"prefix":"10.3390","volume":"19","author":[{"given":"Chunming","family":"Zhang","sequence":"first","affiliation":[{"name":"Department of Statistics, University of Wisconsin-Madison, Madison, WI 53706, USA"}]},{"given":"Zhengjun","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of Wisconsin-Madison, Madison, WI 53706, USA"}]}],"member":"1968","published-online":{"date-parts":[[2017,11,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"43","DOI":"10.2307\/2951475","article-title":"Asymptotics for semiparametric econometric models via stochastic equicontinuity","volume":"62","author":"Andrews","year":"1994","journal-title":"Econometrica"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"931","DOI":"10.2307\/1912705","article-title":"Root-n consistent semiparametric regression","volume":"56","author":"Robinson","year":"1988","journal-title":"Econometrica"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1111\/j.2517-6161.1988.tb01738.x","article-title":"Kernel smoothing in partial linear models","volume":"50","author":"Speckman","year":"1988","journal-title":"J. R. Statist. Soc. B"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1016\/S0165-1765(97)00218-8","article-title":"An elementary estimator of the partial linear model","volume":"57","author":"Yatchew","year":"1997","journal-title":"Econ. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"710","DOI":"10.1198\/016214504000001060","article-title":"New estimation and model selection procedures for semiparametric modeling in longitudinal data analysis","volume":"99","author":"Fan","year":"2004","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"McCullagh, P., and Nelder, J.A. (1989). Generalized Linear Models, Chapman & Hall. [2nd ed.].","DOI":"10.1007\/978-1-4899-3242-6"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1693","DOI":"10.1214\/07-AOS519","article-title":"Semiparametric detection of significant activation for brain fMRI","volume":"36","author":"Zhang","year":"2008","journal-title":"Ann. Stat."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1031","DOI":"10.3150\/bj\/1137421639","article-title":"Profile likelihood inferences on semiparametric varying-coefficient partially linear models","volume":"11","author":"Fan","year":"2005","journal-title":"Bernoulli"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2856","DOI":"10.1214\/009053606000000858","article-title":"Robust estimates in generalized partially linear models","volume":"34","author":"Boente","year":"2006","journal-title":"Ann. Stat."},{"key":"ref_10","first-page":"653","article-title":"Robust-BD estimation and inference for varying-dimensional general linear models","volume":"24","author":"Zhang","year":"2014","journal-title":"Stat. Sin."},{"key":"ref_11","unstructured":"Fan, J., and Gijbels, I. (1996). Local Polynomial Modeling and Its Applications, Chapman and Hall."},{"key":"ref_12","first-page":"620","article-title":"A relaxation method of finding a common point of convex sets and its application to the solution of problems in convex programming","volume":"7","year":"1967","journal-title":"USSR Comput. Math. Math. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Hastie, T., Tibshirani, R., and Friedman, J. (2001). The Elements of Statistical Learning, Springer.","DOI":"10.1007\/978-0-387-21606-5"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1002\/cjs.10005","article-title":"New aspects of Bregman divergence in regression and classification with parametric and nonparametric estimation","volume":"37","author":"Zhang","year":"2009","journal-title":"Can. J. Stat."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1214\/aoms\/1177703732","article-title":"Robust estimation of a location parameter","volume":"35","author":"Huber","year":"1964","journal-title":"Ann. Math. Statist."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Van der Vaart, A.W. (1998). Asymptotic Statistics, Cambridge University Press.","DOI":"10.1017\/CBO9780511802256"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18637\/jss.v033.i01","article-title":"Regularization paths for generalized linear models via coordinate descent","volume":"33","author":"Friedman","year":"2010","journal-title":"J. Stat. Softw."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1080\/01621459.1997.10474001","article-title":"Generalized partially linear single-index models","volume":"92","author":"Carroll","year":"1997","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1080\/01621459.1994.10476447","article-title":"Feasible nonparametric estimation of multiargument monotone functions","volume":"89","author":"Mukarjee","year":"1994","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_20","unstructured":"Albright, S.C., Winston, W.L., and Zappe, C.J. (1999). Data Analysis and Decision Making with Microsoft Excel, Duxbury Press."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"928","DOI":"10.1214\/009053604000000256","article-title":"Nonconcave penalized likelihood with a diverging number of parameters","volume":"32","author":"Fan","year":"2004","journal-title":"Ann. Stat."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1012","DOI":"10.1214\/aos\/1018031266","article-title":"A consistent test for the functional form of a regression based on a difference of variance estimators","volume":"27","author":"Dette","year":"1999","journal-title":"Ann. Stat."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"669","DOI":"10.2307\/3318732","article-title":"Testing additivity by kernel-based methods","volume":"7","author":"Dette","year":"2001","journal-title":"Bernoulli"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1214\/aos\/996986505","article-title":"Generalized likelihood ratio statistics and Wilks phenomenon","volume":"29","author":"Fan","year":"2001","journal-title":"Ann. Stat."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1166","DOI":"10.1214\/13-AOS1099","article-title":"A loss function approach to model specification testing and its relative efficiency","volume":"41","author":"Hong","year":"2013","journal-title":"Ann. Stat."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1016\/0304-4076(95)01760-7","article-title":"A consistent test of functional form via nonparametric estimation techniques","volume":"75","author":"Zheng","year":"1996","journal-title":"J. Econ."},{"key":"ref_27","first-page":"715","article-title":"A root-n consistent backfitting estimator for semiparametric additive modeling","volume":"8","author":"Opsomer","year":"1999","journal-title":"J. Comput. Graph. Stat."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"608","DOI":"10.1093\/restud\/rdt044","article-title":"Inference on treatment effects after selection amongst high-dimensional controls","volume":"81","author":"Belloni","year":"2014","journal-title":"Rev. Econ. Stud."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Cattaneo, M.D., Jansson, M., and Newey, W.K. (2016). Alternative asymptotics and the partially linear model with many regressors. Econ. Theory, 1\u201325.","DOI":"10.1920\/wp.cem.2015.3615"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Cattaneo, M.D., Jansson, M., and Newey, W.K. (arXiv, 2015). Treatment effects with many covariates and heteroskedasticity, arXiv.","DOI":"10.1920\/wp.cem.2015.3715"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1007\/BF00539840","article-title":"Weak and strong uniform consistency of kernel regression estimates","volume":"61","author":"Mack","year":"1982","journal-title":"Z. Wahrsch. Verw. Gebiete"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/11\/625\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:50:27Z","timestamp":1760208627000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/11\/625"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,11,20]]},"references-count":31,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2017,11]]}},"alternative-id":["e19110625"],"URL":"https:\/\/doi.org\/10.3390\/e19110625","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2017,11,20]]}}}