  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).
