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A smooth simultaneous confidence band for correlation curve. (English) Zbl 1404.62048
Summary: A plug-in estimator is proposed for a local measure of variance explained by regression, termed correlation curve in [K. Doksum et al., J. Am. Stat. Assoc. 89, No. 426, 571–582 (1994; Zbl 0803.62035)], 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 [B. W. Hamilton, “Using Engel’s law to estimate CPI Bias”, Am. Econ. Rev. 91, No. 3, 619–630 (2001; doi:10.1257/aer.91.3.619)].

##### MSC:
 62G15 Nonparametric tolerance and confidence regions 62G08 Nonparametric regression and quantile regression 62G20 Asymptotic properties of nonparametric inference 62P20 Applications of statistics to economics 62G07 Density estimation
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