, Volume 27, Issue 2, pp 247–269 | Cite as

A smooth simultaneous confidence band for correlation curve

  • Yuanyuan Zhang
  • Lijian Yang
Original Paper


A plug-in estimator is proposed for a local measure of variance explained by regression, termed correlation curve in Doksum et al. (J Am Stat Assoc 89:571–582, 1994), consisting of a two-step spline–kernel estimator of the conditional variance function and local quadratic estimator of first derivative of the mean function. The estimator is oracally efficient in the sense that it is as efficient as an infeasible correlation estimator with the variance function known. As a consequence of the oracle efficiency, a smooth simultaneous confidence band (SCB) is constructed around the proposed correlation curve estimator and shown to be asymptotically correct. Simulated examples illustrate the versatility of the proposed oracle SCB which confirms the asymptotic theory. Application to a 1995 British Family Expenditure Survey data has found marginally significant evidence for a local version of Engel’s law, i.e., food budget share and household real income are inversely related (Hamilton in Am Econ Rev 91:619–630, 2001).


Confidence band Correlation curve Heteroscedasticity Infeasible estimator Local quadratic estimator 

Mathematics Subject Classification

62G05 62G08 62G10 62G15 62G20 62P20 



This work has been supported in part by 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 two Reviewers, Editor-in-Chief Ana Militino, Prof. Qin Shao, and participants at the First PKU-Tsinghua Colloquium On Statistics for helpful comments.

Supplementary material

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Copyright information

© Sociedad de Estadística e Investigación Operativa 2017

Authors and Affiliations

  1. 1.Center for Statistical Science and Department of Industrial EngineeringTsinghua UniversityBeijingChina

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