Convergence Rates for Kernel Regression in Infinite Dimensional Spaces

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 rates 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 what happens in 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. We also show that 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 of the kernel estimate, and derive the asymptotic behavior of the adaptive estimate.