{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,12]],"date-time":"2026-10-12T21:45:51Z","timestamp":1791841551923,"version":"4.3.4"},"reference-count":28,"publisher":"Institute of Mathematical Statistics","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Ann. Statist."],"published-print":{"date-parts":[[2005,10,1]]},"abstract":"<jats:p>A popular data-driven method for choosing the bandwidth in standard kernel regression is cross-validation. Even when there are outliers in the data, robust kernel regression can be used to estimate the unknown regression curve [Robust and Nonlinear Time Series Analysis. Lecture Notes in Statist. (1984) 26 163\u2013184]. However, under these circumstances standard cross-validation is no longer a satisfactory bandwidth selector because it is unduly influenced by extreme prediction errors caused by the existence of these outliers. A more robust method proposed here is a cross-validation method that discounts the extreme prediction errors. In large samples the robust method chooses consistent bandwidths, and the consistency of the method is practically independent of the form in which extreme prediction errors are discounted. Additionally, evaluation of the method\u2019s finite sample behavior in a simulation demonstrates that the proposed method performs favorably. This method can also be applied to other problems, for example, model selection, that require cross-validation.<\/jats:p>","DOI":"10.1214\/009053605000000499","type":"journal-article","created":{"date-parts":[[2005,11,26]],"date-time":"2005-11-26T03:05:00Z","timestamp":1132974300000},"page":"2291-2310","source":"Crossref","is-referenced-by-count":75,"title":["Cross-validation in nonparametric regression with outliers"],"prefix":"10.1214","volume":"33","author":[{"given":"Denis Heng-Yan","family":"Leung","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"23","doi-asserted-by":"publisher","unstructured":"Rice, J. (1984). Bandwidth choice for nonparametric regression. <i>Ann. Statist.<\/i> <b>12<\/b> 1215--1230.","DOI":"10.1214\/aos\/1176346788"},{"key":"3","doi-asserted-by":"publisher","unstructured":"Cleveland, W. S. (1979). Robust locally weighted regression and smoothing scatterplots. <i>J. Amer. Statist. Assoc.<\/i> <b>74<\/b> 829--836.","DOI":"10.2307\/2286407"},{"key":"1","doi-asserted-by":"publisher","unstructured":"Akaike, H. (1974). A new look at the statistical model identification. <i>IEEE Trans. Automatic Control<\/i> <b>19<\/b> 716--723.","DOI":"10.1109\/TAC.1974.1100705"},{"key":"2","doi-asserted-by":"publisher","unstructured":"Boente, G., Fraiman, R. and Meloche, J. (1997). Robust plug-in bandwidth estimators in nonparametric regression. <i>J. Statist. Plann. Inference<\/i> <b>57<\/b> 109--142.","DOI":"10.1016\/S0378-3758(96)00039-0"},{"key":"4","doi-asserted-by":"publisher","unstructured":"Cox, D. D. (1983). Asymptotics for $M$-type smoothing splines. <i>Ann. Statist.<\/i> <b>11<\/b> 530--551.","DOI":"10.1214\/aos\/1176346159"},{"key":"5","doi-asserted-by":"publisher","unstructured":"Gasser, T. and M\u00fcller, H.-G. (1979). Kernel estimation of regression functions. <i>Smoothing Techniques for Curve Estimation. Lecture Notes in Math.<\/i> <b>757<\/b> 23--68. Springer, Berlin.","DOI":"10.1007\/BFb0098489"},{"key":"6","doi-asserted-by":"publisher","unstructured":"Hall, P., Sheather, S., Jones, M. and Marron, J. (1991). On optimal data-based bandwidth selection in kernel density estimation. <i>Biometrika<\/i> <b>78<\/b> 263--269.","DOI":"10.2307\/2337251"},{"key":"7","doi-asserted-by":"crossref","unstructured":"H\u00e4rdle, W. (1984). How to determine the bandwidth of some nonlinear smoothers in practice. <i>Robust and Nonlinear Time Series Analysis. Lecture Notes in Statist.<\/i> <b>26<\/b> 163--184. Springer, New York.","DOI":"10.1007\/978-1-4615-7821-5_9"},{"key":"8","doi-asserted-by":"crossref","unstructured":"H\u00e4rdle, W. (1987). Algorithm AS 222: Resistant smoothing using the fast Fourier transform. <i>Appl. Statist.<\/i> <b>36<\/b> 104--111.","DOI":"10.2307\/2347850"},{"key":"10","doi-asserted-by":"crossref","unstructured":"H\u00e4rdle, W. and Gasser, T. (1984). Robust non-parametric function fitting. <i>J. Roy. Statist. Soc. Ser. B<\/i> <b>46<\/b> 42--51.","DOI":"10.1111\/j.2517-6161.1984.tb01274.x"},{"key":"11","doi-asserted-by":"publisher","unstructured":"H\u00e4rdle, W., Hall, P. and Marron, J. (1988). How far are automatically chosen regression smoothing parameters from their optimum? (with discussion). <i>J. Amer. Statist. Assoc.<\/i> <b>83<\/b> 86--101.","DOI":"10.2307\/2288922"},{"key":"12","doi-asserted-by":"publisher","unstructured":"H\u00e4rdle, W., Hall, P. and Marron, J. (1992). Regression smoothing parameters that are not far from their optimum. <i>J. Amer. Statist. Assoc.<\/i> <b>87<\/b> 227--233.","DOI":"10.2307\/2290473"},{"key":"13","doi-asserted-by":"publisher","unstructured":"H\u00e4rdle, W. and Luckhaus, S. (1984). Uniform consistency of a class of regression function estimators. <i>Ann. Statist.<\/i> <b>12<\/b> 612--623.","DOI":"10.1214\/aos\/1176346509"},{"key":"16","doi-asserted-by":"publisher","unstructured":"Huber, P. (1964). Robust estimation of a location parameter. <i>Ann. Math. Statist.<\/i> <b>35<\/b> 73--101.","DOI":"10.1214\/aoms\/1177703732"},{"key":"17","doi-asserted-by":"publisher","unstructured":"Leung, D., Marriott, F. and Wu, E. (1993). Bandwidth selection in robust smoothing. <i>J. Nonparametr. Statist.<\/i> <b>2<\/b> 333--339.","DOI":"10.1080\/10485259308832562"},{"key":"20","doi-asserted-by":"crossref","unstructured":"Nadaraya, E. (1964). On estimating regression. <i>Theory Probab. Appl.<\/i> <b>9<\/b> 141--142.","DOI":"10.1137\/1109020"},{"key":"21","doi-asserted-by":"crossref","unstructured":"Park, B. and Marron, J. (1990). Comparison of data-driven bandwidth selectors. <i>J. Amer. Statist. Assoc.<\/i> <b>85<\/b> 66--72.","DOI":"10.1080\/01621459.1990.10475307"},{"key":"22","doi-asserted-by":"crossref","unstructured":"Priestley, M. and Chao, M. (1972). Non-parametric function fitting. <i>J. Roy. Statist. Soc. Ser. B<\/i> <b>34<\/b> 385--392.","DOI":"10.1111\/j.2517-6161.1972.tb00916.x"},{"key":"24","doi-asserted-by":"publisher","unstructured":"Ronchetti, E., Field, C. and Blanchard, W. (1997). Robust linear model selection by cross-validation. <i>J. Amer. Statist. Assoc.<\/i> <b>92<\/b> 1017--1023.","DOI":"10.2307\/2965566"},{"key":"25","doi-asserted-by":"publisher","unstructured":"Shibata, R. (1981). An optimal selection of regression variables. <i>Biometrika<\/i> <b>68<\/b> 45--54.","DOI":"10.2307\/2335804"},{"key":"26","doi-asserted-by":"publisher","unstructured":"Wang, F. and Scott, D. (1994). The $L_1$ method for robust nonparametric regression. <i>J. Amer. Statist. Assoc.<\/i> <b>89<\/b> 65--76.","DOI":"10.2307\/2291201"},{"key":"27","unstructured":"Watson, G. (1964). Smooth regression analysis. <i>Sankhy\u0101 Ser. A<\/i> <b>26<\/b> 359--372."},{"key":"28","doi-asserted-by":"crossref","unstructured":"Whittle, P. (1960). Bounds for the moments of linear and quadratic forms in independent variables. <i>Theory Probab. Appl.<\/i> <b>5<\/b> 302--305.","DOI":"10.1137\/1105028"},{"key":"9","doi-asserted-by":"crossref","unstructured":"H\u00e4rdle, W. (1990). <i>Applied Nonparametric Regression<\/i>. Cambridge Univ. Press.","DOI":"10.1017\/CCOL0521382483"},{"key":"14","unstructured":"H\u00e4rdle, W. and Marron, J. (1983). Optimal bandwidth selection in nonparametric function estimation. Institute of Statistics Mimeo Series no. 1530. Univ. North Carolina, Chapel Hill."},{"key":"15","doi-asserted-by":"crossref","unstructured":"Herrmann, E. (2000). Variance estimation and bandwidth selection for kernel regression. In <i>Smoothing and Regression: Approaches, Computation and Application<\/i> (M. G. Schimek, ed.) 71--107. Wiley, New York.","DOI":"10.1002\/9781118150658.ch4"},{"key":"18","unstructured":"Lietz, A. (2000). <i>Analysis of water-quality trends at two discharge stations---one within Big Cypress National Preserve and one near Biscayne Bay---Southern Florida<\/i>, 1966--1994. U.S. Geological Survey. Available at sofia.usgs.gov\/publications\/wri\/00-4099."},{"key":"19","unstructured":"Marron, J. and H\u00e4rdle, W. (1983). Random approximations to an error criterion of nonparametric statistics. Institute of Statistics Mimeo Series no. 1538. Univ. North Carolina, Chapel Hill."}],"container-title":["The Annals of Statistics"],"original-title":[],"link":[{"URL":"http:\/\/projecteuclid.org\/download\/pdfview_1\/euclid.aos\/1132936564","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,10,12]],"date-time":"2026-10-12T21:00:20Z","timestamp":1791838820000},"score":1,"resource":{"primary":{"URL":"https:\/\/projecteuclid.org\/journals\/annals-of-statistics\/volume-33\/issue-5\/Cross-validation-in-nonparametric-regression-with-outliers\/10.1214\/009053605000000499.full"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,10,1]]},"references-count":28,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2005,10,1]]}},"URL":"https:\/\/doi.org\/10.1214\/009053605000000499","relation":{},"ISSN":["0090-5364","2168-8966"],"issn-type":[{"value":"0090-5364","type":"print"},{"value":"2168-8966","type":"electronic"}],"subject":[],"published":{"date-parts":[[2005,10,1]]}}}