, Volume 24, Issue 3, pp 632–655 | Cite as

A smooth simultaneous confidence band for conditional variance function

  • Li Cai
  • Lijian Yang
Original Paper


A smooth simultaneous confidence band (SCB) is obtained for heteroscedastic variance function in nonparametric regression by applying spline regression to the conditional mean function followed by Nadaraya–Waston estimation using the squared residuals. The variance estimator is uniformly oracally efficient, that is, it is as efficient as, up to order less than \(n^{-1/2}\), the infeasible kernel estimator when the conditional mean function is known, uniformly over the data range. Simulation experiments provide strong evidence that confirms the asymptotic theory while the computing is extremely fast. The proposed SCB has been applied to test for heteroscedasticity in the well-known motorcycle data and Old Faithful geyser data with different conclusions.


B spline Confidence band Heteroscedasticity Infeasible estimator Knots Nadaraya–Waston estimator Variance function 

Mathematics Subject Classification

62G05 62G08 62G10 62G15 62G32 



This work has been supported by NSF award DMS 1007594, Jiangsu Specially-Appointed Professor Program SR10700111, Jiangsu Key-Discipline Program ZY107992, National Natural Science Foundation of China award 11371272, and Research Fund for the Doctoral Program of Higher Education of China award 20133201110002. The authors thank the Editor and two Reviewers for helpful comments.


  1. Akritas MG, Van Keilegom I (2001) ANCOVA methods for heteroscedastic nonparametric regression models. J Am Stat Assoc 96:220–232CrossRefzbMATHGoogle Scholar
  2. Bickel PJ, Rosenblatt M (1973) On some global measures of deviations of density function estimates. Ann Stat 31:1852–1884MathSciNetGoogle Scholar
  3. Brown DL, Levine M (2007) variance estimation in nonparametric regression via the difference sequence method. Ann Stat 35:2219–2232MathSciNetCrossRefzbMATHGoogle Scholar
  4. Cai T, Wang L (2008) Adaptive variance function estimation in heteroscedastic nonparametric regression. Ann Stat 36:2025–2054MathSciNetCrossRefzbMATHGoogle Scholar
  5. Carroll RJ, Wang Y (2008) Nonparametric variance estimation in the analysis of microarray data: a measurement error approach. Biometrika 95:437–449MathSciNetCrossRefzbMATHGoogle Scholar
  6. Carroll RJ, Ruppert D (1988) Transformations and weighting in regression. Champman and Hall, LondonCrossRefGoogle Scholar
  7. Claeskens G, Van Keilegom I (2003) Bootstrap confidence bands for regression curves and their derivatives. Ann Stat 31:1852–1884CrossRefzbMATHGoogle Scholar
  8. Davidian M, Carroll RJ, Smith W (1988) Variance functions and the minimum detectable concentration in assays. Biometrika 75:549–556MathSciNetCrossRefzbMATHGoogle Scholar
  9. De Boor C (2001) A practical guide to splines. Springer, New YorkzbMATHGoogle Scholar
  10. Dette H, Munk A (1998) Testing heteroscedasticity in nonparametric regression. J R Stat Soc Ser B 60:693–708MathSciNetCrossRefzbMATHGoogle Scholar
  11. Fan J, Gijbels T (1996) Local polynomial modelling and its applications. Champman and Hall, LondonzbMATHGoogle Scholar
  12. Fan J, Yao Q (1998) Efficient estimation of conditional variance functions in stochastic regression. Biometrika 85:645–660Google Scholar
  13. Hall P, Titterington MD (1988) On confidence bands in nonparametric density estimation and regression. J Multivar Anal 27:228–254Google Scholar
  14. Hall P, Carroll RJ (1989) Variance function estimation in regression: the effect of estimating the mean. J R Stat Soc Ser B 51:3–14MathSciNetzbMATHGoogle Scholar
  15. Hall P, Marron JS (1990) On variance estimation in nonparametric regression. Biometrika 77:415–419MathSciNetCrossRefzbMATHGoogle Scholar
  16. Härdle W (1989) Asmptotic maximal deviation of M-smoothers. J Multivar Anal 29:163–179CrossRefzbMATHGoogle Scholar
  17. Härdle W (1992) Applied nonparametric regression. Cambridge University Press, CambridgeGoogle Scholar
  18. Levine M (2006) Bandwidth selection for a class of difference-based variance estimators in the nonparametric regression: a possible approach. Comput Stat Data Anal 50:3405–3431CrossRefzbMATHGoogle Scholar
  19. Liu R, Yang L, Härdle W (2013) Oracally efficient two-step estimation of generalized additive model. J Am Stat Assoc 108:619–631Google Scholar
  20. Ma S, Yang L, Carroll RJ (2012) A simultaneous confidence band for sparse longitudinal regression. Stat Sin 22:95–122MathSciNetzbMATHGoogle Scholar
  21. Müller HG, Stadtmüller U (1987) Estimation of heteroscedasticity in regression analysis. Ann Stat 15:610–625CrossRefzbMATHGoogle Scholar
  22. Silverman WB (1986) Density estimation for statistics and data analysis. Chapman and Hall, LondonCrossRefzbMATHGoogle Scholar
  23. Song Q, Yang L (2009) Spline confidence bands for variance functions. J Nonparametr Stat 5:589–609MathSciNetCrossRefGoogle Scholar
  24. Tusnády G (1977) A remark on the approximation of the sample df in the multidimensional case. Periodica Mathematica Hungarica 8:53–55MathSciNetCrossRefzbMATHGoogle Scholar
  25. Wang L, Brown LD, Cai T, Levine M (2008) Effect of mean on variance function estimation in nonparametric regression. Ann Stat 36:646–664MathSciNetCrossRefzbMATHGoogle Scholar
  26. Wang J, Liu R, Cheng F, Yang L (2014) Oracally efficient estimation of autoregressive error distribution with simultaneous confidence band. Ann Stat 42:654–668MathSciNetCrossRefzbMATHGoogle Scholar
  27. Wang L, Yang L (2007) Spline-backfitted kernel smoothing of nonlinear additive autoregression model. Ann Stat 35:2474–2503CrossRefzbMATHGoogle Scholar
  28. Wang J, Yang L (2009a) Polynomial spline confidence bands for regression curves. Stat Sin 19:325–342zbMATHGoogle Scholar
  29. Wang L, Yang L (2009b) Spline estimation of single-index models. Stat Sin 19:765–783zbMATHGoogle Scholar
  30. Wang J, Yang L (2009c) Efficient and fast spline-backfitted kernel smoothing of additive models. Ann Inst Stat Math 61:663–690CrossRefzbMATHGoogle Scholar
  31. Xia Y (1998) Bias-corrected confidence bands in nonparametric regression. J R Stat Soc Ser B 60:797–811CrossRefzbMATHGoogle Scholar
  32. Xue L, Yang L (2006) Additive coefficient modeling via polynomial spline. Stat Sin 16:1423–1446MathSciNetGoogle Scholar
  33. Zheng S, Yang L, Härdle W (2014) A smooth simultaneous confidence corridor for the mean of sparse functional data. J Am Stat Assoc 109:661–673CrossRefGoogle Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2015

Authors and Affiliations

  1. 1.Center for Advanced Statistics and Econometrics ResearchSoochow UniversitySuzhouChina

Personalised recommendations