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

A smooth simultaneous confidence band for conditional variance function

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.


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Copyright information

© Sociedad de Estadística e Investigación Operativa 2015

Authors and Affiliations

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

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