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Convergence rates for kernel regression in infinite-dimensional spaces

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  • Joydeep Chowdhury

    (Indian Statistical Institute)

  • Probal Chaudhuri

    (Indian Statistical Institute)

Abstract

We consider a nonparametric regression setup, where the covariate is a random element in a complete separable metric space, and the parameter of interest associated with the conditional distribution of the response lies in a separable Banach space. We derive the optimum convergence rate for the kernel estimate of the parameter in this setup. The small ball probability in the covariate space plays a critical role in determining the asymptotic variance of kernel estimates. Unlike the case of finite-dimensional covariates, we show that the asymptotic orders of the bias and the variance of the estimate achieving the optimum convergence rate may be different for infinite-dimensional covariates. Also, the bandwidth, which balances the bias and the variance, may lead to an estimate with suboptimal mean square error for infinite-dimensional covariates. We describe a data-driven adaptive choice of the bandwidth and derive the asymptotic behavior of the adaptive estimate.

Suggested Citation

  • Joydeep Chowdhury & Probal Chaudhuri, 2020. "Convergence rates for kernel regression in infinite-dimensional spaces," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(2), pages 471-509, April.
    • Handle: RePEc:spr:aistmt:v:72:y:2020:i:2:d:10.1007_s10463-018-0697-2
    DOI: 10.1007/s10463-018-0697-2
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    References listed on IDEAS

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