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Semi-parametric estimation of partially linear single-index models. (English) Zbl 1089.62050
Summary: One of the most difficult problems in applications of semi-parametric partially linear single-index models (PLSIM) is the choice of pilot estimators and complexity parameters which may result in radically different estimators. Pilot estimators are often assumed to be root-\(n\) consistent, although they are not given in a constructible way. Complexity parameters, such as a smoothing bandwidth, are constrained to a certain speed, which is rarely determinable in practical situations.
In this paper, efficient, constructible and practicable estimators of PLSIMs are designed with applications to time series. The proposed technique answers two questions of R. J. Carroll et al. [Generalized partially linear single-index models. J. Am. Stat. Assoc. 92, No. 438, 477–489 (1997; Zbl 0890.62053)]: no root-\(n\) pilot estimator for the single-index part of the model is needed and complexity parameters can be selected at the optimal smoothing rate. The asymptotic distribution is derived and the corresponding algorithm is easily implemented. Examples from real data sets (credit-scoring and environmental statistics) illustrate the technique and the proposed methodology of minimum average variance estimation (MAVE).

MSC:
62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G07 Density estimation
62F12 Asymptotic properties of parametric estimators
62E20 Asymptotic distribution theory in statistics
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