Calibrating the degrees of freedom for automatic data smoothing and effective curve checking.

*(English)*Zbl 1040.62027Summary: Curve fitting and curve checking based on the local polynomial regression technique are commonly used data-analytic methods in statistics. This article examines, in nonparametric settings, both the asymptotic expressions and empirical formulas for degrees of freedom (DF), a notion introduced by T. J. Hastie and R. J. Tibshirani [Generalized additive models. (1990; Zbl 0747.62061); J. R. Stat. Soc., Ser. B 55, 757–796 (1993; Zbl 0796.62060)], of linear smoothers. The asymptotic results give useful insights into the nonparametric modeling complexity. Meanwhile, by substituting the exact DFs by the empirical formula, an empirical version of the generalized cross-validation (EGCV) is obtained.

An automatic bandwidth selection method based on minimizing EGCV is proposed for conducting local smoothing. This procedure preserves full benefits of the ordinary and generalized cross-validation, but offers a substantial reduction in computational burden. Furthermore, the EGCV-minimizing bandwidth can be extended in a very simple manner to fit multivariate models, such as the varying-coefficient models. Applications of calibrating DFs to important inferential issues, such as assessing the validity of useful model assumptions and measuring the significance of predictor variables based on the generalized likelihood ratio statistics are also discussed. Simulation studies are presented to illustrate the performance of the proposed procedures in a range of statistical problems.

An automatic bandwidth selection method based on minimizing EGCV is proposed for conducting local smoothing. This procedure preserves full benefits of the ordinary and generalized cross-validation, but offers a substantial reduction in computational burden. Furthermore, the EGCV-minimizing bandwidth can be extended in a very simple manner to fit multivariate models, such as the varying-coefficient models. Applications of calibrating DFs to important inferential issues, such as assessing the validity of useful model assumptions and measuring the significance of predictor variables based on the generalized likelihood ratio statistics are also discussed. Simulation studies are presented to illustrate the performance of the proposed procedures in a range of statistical problems.

##### MSC:

62G07 | Density estimation |

62G20 | Asymptotic properties of nonparametric inference |

62G08 | Nonparametric regression and quantile regression |