Convergence rates of empirical block length selectors for block bootstrap

We investigate the accuracy of two general non-parametric methods for estimating optimal block lengths for block bootstraps with time series - the first proposed in the seminal paper of Hall, Horowitz and Jing (Biometrika 82 (1995) 561-574) and the second from Lahiri et al. (Stat. Methodol. 4 (2007) 292-321). The relative performances of these general methods have been unknown and, to provide a comparison, we focus on rates of convergence for these block length selectors for the moving block bootstrap (MBB) with variance estimation problems under the smooth function model. It is shown that, with suitable choice of tuning parameters, the optimal convergence rate of the first method is $O_p(n^{-1/6})$ where $n$ denotes the sample size. The optimal convergence rate of the second method, with the same number of tuning parameters, is shown to be $O_p(n^{-2/7})$, suggesting that the second method may generally have better large-sample properties for block selection in block bootstrap applications beyond variance estimation. We also compare the two general methods with other plug-in methods specifically designed for block selection in variance estimation, where the best possible convergence rate is shown to be $O_p(n^{-1/3})$ and achieved by a method from Politis and White (Econometric Rev. 23 (2004) 53-70).