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Minimax rate of convergence for an estimator of the functional component in a semiparametric multivariate partially linear model

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  • Levine, Michael

Abstract

A multivariate semiparametric partial linear model for both fixed and random design cases is considered. Earlier, in Brown et al. (2014), the model has been analyzed using a difference sequence approach. In particular, the functional component has been estimated using a multivariate Nadaraya–Watson kernel smoother of the residuals of the linear fit. Moreover, this functional component estimator has been shown to be rate optimal if the Lipschitz smoothness index exceeds half the dimensionality of the functional component domain. In the current manuscript, we take this research further and show that, for both fixed and random designs, the rate achieved is the minimax rate under both risk at a point and the L2 risk. The result is achieved by proving lower bounds on both pointwise risk and the L2 risk of possible estimators of the functional component.

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  • Levine, Michael, 2015. "Minimax rate of convergence for an estimator of the functional component in a semiparametric multivariate partially linear model," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 283-290.
    • Handle: RePEc:eee:jmvana:v:140:y:2015:i:c:p:283-290
    DOI: 10.1016/j.jmva.2015.05.010
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    References listed on IDEAS

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    1. Cai, T. Tony & Levine, Michael & Wang, Lie, 2009. "Variance function estimation in multivariate nonparametric regression with fixed design," Journal of Multivariate Analysis, Elsevier, vol. 100(1), pages 126-136, January.
    2. Axel Munk & Nicolai Bissantz & Thorsten Wagner & Gudrun Freitag, 2005. "On difference‐based variance estimation in nonparametric regression when the covariate is high dimensional," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(1), pages 19-41, February.
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    Cited by:

    1. Yuejin Zhou & Yebin Cheng & Wenlin Dai & Tiejun Tong, 2018. "Optimal difference-based estimation for partially linear models," Computational Statistics, Springer, vol. 33(2), pages 863-885, June.
    2. Michael Levine, 2019. "Robust functional estimation in the multivariate partial linear model," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(4), pages 743-770, August.

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