Convergence rates for kernel regression in infinite-dimensional spaces

Joydeep Chowdhury () and Probal Chaudhuri ()
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Joydeep Chowdhury: Indian Statistical Institute
Probal Chaudhuri: Indian Statistical Institute

Annals of the Institute of Statistical Mathematics, 2020, vol. 72, issue 2, No 6, 509 pages

Abstract: 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.

Keywords: Adaptive estimate; Bias-variance decomposition; Gaussian process; Maximum likelihood regression; Mean square error; Optimal bandwidth; Small ball probability; t process (search for similar items in EconPapers)
Date: 2020
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Citations: View citations in EconPapers (5)

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DOI: 10.1007/s10463-018-0697-2

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