blocklength 0.2.0
R-bloggers 2025-02-18
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I’m excited to announce that blocklength 0.2.0 is now available on CRAN! blocklength is designed to be used with block-bootstrap procedures and makes it quick and easy to select a block-length quantitatively. This significant update includes a new block-length selection algorithm by Lahiri, Furukawa, and Lee (2007), the nonparametric plug-in “NPPI” method.
Install it with:
install.packages("blocklength")The most important changes are highlighted below and you can see a full list of changes in the changelog.
New Features
The new nppi() function and its corresponding S3 plot method plot.nppi() can be used in a similar manner to other algorithms already present in the library. Documentation can be found at the package website. The NPPI method brings additional flexibility for users, and extends the usefulness of the package to a wider-range of estimators including bias, variance, distribution function, and quantile estimators.
The NPPI Algorithm
The NPPI algorithm is based on theoretical foundations first described in the seminal paper by Hall, Horowitz, and Jing (1995) “HHJ”, who proposed the first optimal block-length selection algorithm for the block-bootstrap. Their hhj() algorithm has been part of blocklength since the beginning, but the NPPI method both relaxes the assumptions of the HHJ method and extends it to a wider range of estimators.
HHJ show that for many block bootstrap estimators, the variance of the bootstrap estimator is an increasing function of the block-length \(\ell\) while its bias is a decreasing function of \(\ell\). In equations (2.1) and (2.2), Lahiri, Furukawa, and Lee (2007) show that under suitable regularity conditions, the bias and variance estimators can be expanded such that:
\[n^{2a} \cdot \text{Var}(\hat{\varphi}_n(\ell)) = C_1 n^{-1} \ell^r + o(n^{-1} \ell^r) \quad \text{as } n \to \infty\]
\[n^a \cdot \text{Bias}(\hat{\varphi}_n(\ell)) = C_2 \ell^{-1} + o(\ell^{-1}) \quad \text{as } n \to \infty\]
where \(C_1\) and \(C_2\) are population parameters.
The key finding from Lahiri, Furukawa, and Lee (2007) is that rather than deriving analytical expressions for these population constants, we can use nonparametric resampling methods (i.e. the moving block-bootstrap and the moving-blocks-jackknife) to estimate them directly. Moreover, these estimators are shown to be consistent under the regularity conditions of HHJ.
Putting this all together, NPPI has the following three steps:
- Compute the bias estimator using the moving block-bootstrap (MBB).
- Use the bootstrap replicates (blocks) from (1) to compute the variance estimator using the moving-blocks-jackknife (MBJ).
- Compute the population parameter estimates \(\hat{C_1}\) and \(\hat{C_2}\) and the final estimator for the optimal block-length.
Step 1
Bias estimation is done via a very neat trick. We simply compute the block-bootstrap for a chosen block-length \(\ell\) and then \(2\ell\). The bias estimator is then simply given by equation (3.9):
\[\widehat{\text{BIAS}}_n \equiv \widehat{\text{BIAS}}_n(\ell) = 2 \left( \hat{\varphi}_n(\ell) – \hat{\varphi}_n(2\ell) \right).\]
A diagram of the MBB estimator \(\hat{\varphi}_n\) is given in Lahiri (2003) Resampling methods for dependent data, Figure 7.1:
Step 2
Variance estimation is computed using the moving-blocks-jackknife (MBJ) method of Liu and Singh (1992). The MBJ method is a generalization of the jackknife-after-bootstrap method of Efron (1992) to the block-bootstrap. The MBJ method is a very clever way to estimate the variance of a block-bootstrap estimator because we can simply regroup the resampled blocks from Step 1 without recomputing the entire block-bootstrap for each iteration.
For each of of the resampled blocks-sets created from the MBB, we first delete \(m\) consecutive blocks (this set of blocks is referred to as the block-of-blocks), resample the remaining blocks randomly with replacement, and compute the \(i^{th}\) jackknife point value \(\hat{\varphi}^{(i)}_n\) by re-using the MBB estimator \(\hat{\varphi}_n\). We calculate the JAB point values a total of \(M\) times and then compute the JAB variance estimator following equation (3.6):
\[\widehat{\text{VAR}}_{\text{JAB}}(\hat{\varphi}_n) =\frac{m}{(N – m)} \frac{1}{M} \sum_{i=1}^{M} \left( \tilde{\varphi}_n^{(i)} – \hat{\varphi}_n \right)^2\]
where \(\tilde{\varphi}_n^{(i)} = m^{-1} \left[ N \hat{\varphi}_n – (N – m) \hat{\varphi}_n^{(i)} \right]\).
A diagram of the JAB point value computation is given by Lahiri (2003), Figure 7.2:
Step 3
The final step is to compute the NPPI estimator of the optimal block-length \(\hat{\ell}^0\), given by equation (4.15):
\[\hat{\ell}^0 = \left[ \frac{2 \hat{C}_2^2}{r \hat{C}_1} \right]^{\frac{1}{r+2}} n^{\frac{1}{r+2}}\]where \(\hat{C}_1 = n \ell^{-r} \widehat{\text{VAR}}\) and \(\hat{C}_2 = \widehat{\text{BIAS}}_n\).
Conclusion
Implementing the NPPI method was essentially a very fun exercise in meta-resampling. It makes use of the two most popular resampling methods, the bootstrap and the jackknife, as a means to estimate parameters for the block-bootstrap: itself a resampling algorithm. The NPPI method is a great addition to blocklength and I hope it will be useful to researchers and practitioners alike!
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