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Original Articles

Convergence rates for uniform confidence intervals based on local polynomial regression estimators

, &
Pages 31-48 | Received 03 Dec 2014, Accepted 11 Oct 2015, Published online: 26 Nov 2015

References

  • Bernardo, J.M., and Smith, A.F.M. (2000), Bayesian Theory, Chichester: John Wiley.
  • Bickel, P.J., and Rosenblatt, M. (1973), ‘On Some Global Measures of the Deviations of Density Function Estimates’, The Annals of Statistics, 1, 1071–1095. doi: 10.1214/aos/1176342558
  • Cabrera, J. (2012), locpol: Kernel Local Polynomial Regression, R package version 0.6-0. http://cran.r-project.org/web/packages/locpol/index.htmlhttp://cran.r-project.org/web/packages/locpol/index.html.
  • Chaudhuri, P., and Marron, J.S. (1999), ‘SiZer for Exploration of Structures in Curves’, Journal of the American Statistical Association, 94, 807–823. doi: 10.1080/01621459.1999.10474186
  • Chen, S.X., and Qin, Y.S. (2000), ‘Empirical Likelihood Confidence Intervals for Local Linear Smoothers’, Biometrika, 87, 946–953. doi: 10.1093/biomet/87.4.946
  • Chen, S.X., and Qin, Y.S. (2002), ‘Confidence Intervals Based on Local Linear Smoother’, Scandinavian Journal of Statistics, 29, 89–99. doi: 10.1111/1467-9469.00273
  • Claeskens, G., and Van Keilegom, I. (2003), ‘Bootstrap Confidence Bands for Regression Curves and Their Derivatives’, The Annals of Statistics, 31, 1852–1884. doi: 10.1214/aos/1074290329
  • Cleveland, W.S. (1979), ‘Robust Locally Weighted Regression and Smoothing Scatterplots’, Journal of the American Statistical Association, 74, 829–836. doi: 10.1080/01621459.1979.10481038
  • De Brabanter, K., Ferrario, P.G., and Györfi, L. (2014), ‘Detecting Ineffective Features for Nonparametric Regression,’ in Regularization, Optimization, Kernels, and Support Vector Machines, eds. J.A.K. Suykens, M. Signoretto, and A. Argyriou, Machine Learning & Pattern Recognition, Leuven: Chapman & Hall, Chapter 8, pp. 177–194.
  • Devroye, L., Ferrario, P.G., Györfi, L., and Walk, H. (2013), ‘Strong Universal Consistent Estimate of the Minimum Mean Squared Error,’ in Empirical Inference – Festschrift in Honor of Vladimir N. Vapnik, eds. B. Schölkopf, Z. Luo, and V. Vovk, Tübingen: Springer, Chapter 14, pp. 143–160.
  • Eubank, R.L., and Speckman, P.L. (1993), ‘Confidence Bands in Nonparametric Regression’, Journal of the American Statistical Association, 88, 1287–1301. doi: 10.1080/01621459.1993.10476410
  • Fan, J. (1992), ‘Design-Adaptive Nonparametric Regression’, Journal of the American Statistical Association, 87, 998–1004. doi: 10.1080/01621459.1992.10476255
  • Fan, J. (1993), ‘Local Linear Regression Smoothers and their Minimax Efficiency’, The Annals of Statistics, 21, 196–216. doi: 10.1214/aos/1176349022
  • Fan, J., and Gijbels, I. (1992), ‘Variable Bandwidth and Local Linear Regression Smoothers’, The Annals of Statistics, 20, 2008–2036. doi: 10.1214/aos/1176348900
  • Fan, J., and Gijbels, I. (1995), ‘Data-Driven Bandwidth Selection in Local Polynomial Fitting: Variable Bandwidth and Spatial Adaptation’, Journal of the Royal Statistical Society B, 57, 371–394.
  • Fan, J., and Gijbels, I. (1996), Local Polynomial Modeling and Its Applications, London: Chapman & Hall.
  • Hall, P. (1991), ‘On Convergence Rates of Suprema’, Probability Theory and Related Fields, 89, 447–455. doi: 10.1007/BF01199788
  • Hall, P. (1992), ‘On Bootstrap Confidence Intervals in Nonparametric Regression’, The Annals of Statistics, 20, 695–711. doi: 10.1214/aos/1176348652
  • Hall, P., Kay, J.W., and Titterington, D.M. (1990), ‘Asymptotically Optimal Difference-Based Estimation of Variance in Nonparametric Regression’, Biometrika, 77, 521–528. doi: 10.1093/biomet/77.3.521
  • Hall, P., and Titterington, D.M. (1988), ‘On Confidence Bands in Nonparametric Density Estimation and Regression’, Journal of Multivariate Analysis, 27, 228–254. doi: 10.1016/0047-259X(88)90127-3
  • Härdle, W. (1989), ‘Asymptotic Maximal Deviation of M-Smoothers’, Journal of Multivariate Analysis, 29, 163–179. doi: 10.1016/0047-259X(89)90022-5
  • Härdle, W., Hall, P., and Marron, J.S. (1992), ‘Regression Smoothing Parameters that are not Far from their Optimum’, Journal of the American Statistical Association, 87, 227–233.
  • Härdle, W., and Marron, J.S. (1991), ‘Bootstrap Simultaneous Error Bars for Nonparametric Regression’, The Annals of Statistics, 19, 778–796. doi: 10.1214/aos/1176348120
  • Hotelling, H. (1939), ‘Tubes and Spheres in n-Spaces, and a Class of Statistical Problems’, American Journal of Mathematics, 61, 440–460. doi: 10.2307/2371512
  • Huang, L.S. (1995), On Nonparametric Estimation and Goodness-of-Fit, Chapel Hill, NC: The University of North Carolina.
  • Knafl, G., Sacks, J., and Ylvisaker, D. (1985), ‘Confidence Bands for Regression Functions’, Journal of the American Statistical Association, 80, 683–691. doi: 10.1080/01621459.1985.10478169
  • Krivobokova, T., Kneib, T., and Claeskens, G. (2010), ‘Simultaneous Confidence Bands for Penalized Spline Estimators’, Journal of the American Statistical Association, 105, 852–863. doi: 10.1198/jasa.2010.tm09165
  • Loader, C. (1993), ‘Nonparametric Regression, Confidence Bands and Bias Correction.’ in Proceedings of the 25th Symposium on the Interface Between Computer Science and Statistics, pp. 131–136.
  • Loader, C., and Sun, J. (1997), ‘Robustness of Tube Formula Based Confidence Bands’, Journal of Computational and Graphical Statistics, 6, 242–250.
  • Müller, H. (1985), ‘Empirical Bandwidth Choice for Nonparametric Kernel Regression by Means of Pilot Estimators’, Statistical Decisions, 2, 193–206.
  • Nadaraya, E.A. (1964), ‘On Estimating Regression’, Theory of Probability and its Applications, 9, 141–142. doi: 10.1137/1109020
  • Naiman, D.Q. (1986), ‘Conservative Confidence Bands in Curvilinear Regression’, The Annals of Statistics, 14, 896–906. doi: 10.1214/aos/1176350040
  • Naiman, D.Q. (1990), ‘On Volumes of Tubular Neighborhoods of Spherical Polyhedra and Statistical Inference’, The Annals of Statistics, 18, 685–716. doi: 10.1214/aos/1176347621
  • Neumann, M.H. (1995), ‘Automatic Bandwidth Choice and Confidence Intervals in Nonparametric Regression’, The Annals of Statistics, 23, 1937–1959. doi: 10.1214/aos/1034713641
  • Neumann, M.H., and Polzehl, J. (1998), ‘Simultaneous Bootstrap Confidence Bands in Nonparametric Regression’, Journal of Nonparametric Statistics, 9, 307–333. doi: 10.1080/10485259808832748
  • Oehlert, G.W. (1993), ‘Regional Trends in Sulfate Wet Deposition’, Journal of the American Statistical Association, 88, 390–399. doi: 10.1080/01621459.1993.10476288
  • Rice, S.O. (1939), ‘The Distribution of the Maxima of a Random Curve’, American Journal of Mathematics, 61, 409–416. doi: 10.2307/2371510
  • Ruppert, D. (1995), ‘Empirical-Bias Bandwidths for Local Polynomial Nonparametric Regression and Density Estimation’, Journal of the American Statistical Association, 92, 1049–1062. doi: 10.1080/01621459.1997.10474061
  • Ruppert, D., and Wand, M.P. (1994), ‘Multivariate Locally Weighted Least Squares Regression’, The Annals of Statistics, 22, 1346–1370. doi: 10.1214/aos/1176325632
  • Ruppert, D., Wand, M.P., and Carroll, R.J. (2003), Semiparametric Regression, Cambridge: Cambridge University Press.
  • Silverman, B. (1986), Density Estimation for Statistics and Data Analysis, London: Chapman & Hall.
  • Stone, C.J. (1977), ‘Consistent Nonparametric Regression’, The Annals of Statistics, 5, 595–645. doi: 10.1214/aos/1176343886
  • Stone, C.J. (1980), ‘Optimal Rates of Convergence for Nonparametric Estimators’, The Annals of Statistics, 8, 1348–1360. doi: 10.1214/aos/1176345206
  • Sun, J. (1993), ‘Probabilities of the Maxima of Gaussian Random Fields’, The Annals of Probability, 21, 852–855. doi: 10.1214/aop/1176989393
  • Sun, J., and Loader, C.R. (1994), ‘Simultaneous Confidence Bands for Linear Regression and Smoothing’, The Annals of Statistics, 22, 1328–1345. doi: 10.1214/aos/1176325631
  • Ullah, A. (1985), ‘Specification Analysis of Econometric Models’, Journal of Quantitative Economics, 2, 187–209.
  • Wahba, G. (1983), ‘Bayesian Confidence Intervals for the Cross-Validated Smoothing Spline’, Journal of the Royal Statistical Society, Series B, 45, 133–150.
  • Watson, G.S. (1964), ‘Smooth Regression Analysis’, Sankhyā: The Indian Journal of Statistics, Series A, 26, 359–372.
  • Weyl, H. (1939), ‘On the Volume of Tubes’, American Journal of Mathematics, 61, 461–472. doi: 10.2307/2371513
  • Wiesenfarth, M., Krivobokova, T., Klasen, S., and Sperlich, S. (2012), ‘Direct Simultaneous Inference in Additive Models and Its Application to Model Undernutrition’, Journal of the American Statistical Association, 107, 1286–1296. doi: 10.1080/01621459.2012.682809
  • Xia, Y. (1998), ‘Bias-Corrected Confidence Bands in Nonparametric Regression’, Journal of the Royal Statistical Society, Series B, 60, 797–811. doi: 10.1111/1467-9868.00155
  • Xia, Y., and Li, W.K. (2002), ‘Asymptotic Behavior of Bandwidth Selected by the Cross-Validation Method for Local Polynomial Fitting’, Journal of Multivariate Analysis, 83, 265–287. doi: 10.1006/jmva.2001.2048

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