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

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 L2L2 risk. The result is achieved by proving lower bounds on both pointwise risk and the L2L2 risk of possible estimators of the functional component.