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Variance estimation is an important aspect in statistical inference, especially in the dependent data situations. Resamplingmethods are ideal for solving this problem since these do not require restrictive distributional assumptions. In this paper, wedevelop a novel resampling method in the Jackknife family called the stationary jackknife. It can be used to estimatethe variance of a statistic in the cases where observations are from a general stationary sequence. Unlike the moving blockjackknife, the stationary jackknife computes the jackknife replication by deleting a variable length block and thelength has a truncated geometric distribution. Under appropriate assumptions, we can show the stationary jackknifevariance estimator is a consistent estimator for the case of the sample mean and, more generally, for a class of nonlinearstatistics. Further, the stationary jackknife is shown to provide reasonable variance estimation for a wider range ofexpected block lengths when compared with the moving block jackknife by simulation.
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Bias, Consistency, and Alternative Perspectives of the Infinitesimal Jackknife
Though introduced nearly 50 years ago, the infinitesimal jackknife (IJ) remains a popular modern tool for quantifying predictive uncertainty in complex estimation settings. In particular, when supervised learning ensembles are constructed via bootstrap samples, recent work demonstrated that the IJ estimate of variance is particularly convenient and useful. However, despite the algebraic simplicity of its final form, its derivation is rather complex. As a result, studies clarifying the intuition behind the estimator or rigorously investigating its properties have been severely lacking. This work aims to take a step forward on both fronts. We demonstrate that surprisingly, the exact form of the IJ estimator can be obtained via a straightforward linear regression of the individual bootstrap estimates on their respective weights or via the classical jackknife. The latter realization allows us to formally investigate the bias of the IJ variance estimator and better characterize the settings in which its use is appropriate. Finally, we extend these results to the case of U-statistics where base models are constructed via subsampling rather than bootstrapping and provide a consistent estimate of the resulting variance.
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- Award ID(s):
- 2015400
- PAR ID:
- 10536191
- Editor(s):
- Chen, Yi-Hau; Stufken, John; Judy_Wang, Huixia
- Publisher / Repository:
- Statistica Sinica
- Date Published:
- Journal Name:
- Statistica Sinica
- ISSN:
- 1017-0405
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.5705/ss.202022.0131
