Optimal uniform convergence rates and asymptotic normality for series estimators under weak dependence and weak conditions - Xiaohong Chen; Timothy M. Christensen (Centre for Microdata Methods and Practice (cemmap), 2014)
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.
Keywords: C12; C14; C32; 330; Nonparametric series regression; Optimal uniform convergence rates; Weak dependence; Random matrices; Splines; Wavelets; (Nonlinear) Irregular Functionals; Sieve t statistics