Optimal Uniform Convergence Rates and Asymptotic Normality for Series Estimators Under Weak Dependence and Weak Conditions

TL;DR

This paper demonstrates that spline and wavelet series regression estimators achieve optimal uniform convergence rates under weak dependence and conditions, and establishes asymptotic normality of t statistics for complex functionals.

Contribution

It proves optimal convergence rates for series estimators under weak dependence and heavy-tailed errors, and introduces a new exponential inequality for weakly dependent random matrices.

Findings

Achieves optimal uniform convergence rate $(n/ ext{log} n)^{-p/(2p+d)}$

Establishes asymptotic normality for t statistics of nonlinear functionals

Develops a new exponential inequality for sums of weakly dependent random matrices

Abstract

We show that spline and wavelet series regression estimators for weakly dependent regressors attain the optimal uniform (i.e. sup-norm) convergence rate $(n/\log n)^{-p/(2p+d)}$ of Stone (1982), where $d$ is the number of regressors and $p$ is the smoothness of the regression function. The optimal rate is achieved even for heavy-tailed martingale difference errors with finite $(2+(d/p))$th absolute moment for $d/p<2$. We also establish the asymptotic normality of t statistics for possibly nonlinear, irregular functionals of the conditional mean function under weak conditions. The results are proved by deriving a new exponential inequality for sums of weakly dependent random matrices, which is of independent interest.