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. 2020;29(3):608-619.
doi: 10.1080/10618600.2020.1714633. Epub 2020 Mar 12.

The automatic construction of bootstrap confidence intervals

Bradley Efron  1 , Balasubramanian Narasimhan  1
Affiliations

The automatic construction of bootstrap confidence intervals

Bradley Efron et al. J Comput Graph Stat. 2020.

Abstract

The standard intervals, e.g., θ ^ ± 1.96 σ ^ for nominal 95% two-sided coverage, are familiar and easy to use, but can be of dubious accuracy in regular practice. Bootstrap confidence intervals offer an order of magnitude improvement-from first order to second order accuracy. This paper introduces a new set of algorithms that automate the construction of bootstrap intervals, substituting computer power for the need to individually program particular applications. The algorithms are described in terms of the underlying theory that motivates them, along with examples of their application. They are implemented in the R package bcaboot.

Keywords: bca method; exponential families; nonparametric intervals; second-order accuracy.

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Figures

Fig. 1
Fig. 1
Two-sided nonparametric confidence limits for adjusted R2, diabetes data, plotted vertically versus nominal converage level. Bootstrap (solid curves); standard intervals (dotted curves). Small red vertical bars indicate Monte Carlo error in the bootstrap curves, from program bcajack (Section 3). Horizontal dashed line shows θ^=0.507, seen to lie closer to upper bootstrap limits than to lower ones.
Fig. 2
Fig. 2
2000 nonparametric bootstrap replications of the adjusted R2 statistic for the diabetes example of Figure 1. Only 40.2% of the 2000 θ^* values were less than θ^=0.507, suggesting its strong upward bias.
Fig. 3
Fig. 3
Neonate data. Left panel: 2000 parametric bootstrap replications resp*, MLE. Right panel: bca limits (solid) compared to standard limits (dotted), from bcapar. Estimates z^0=−0.215, a^=−0.019.
Fig. 4
Fig. 4
As in 3, but now using glmnet estimation for resp, rather than logistic regression MLE.
Fig. 5
Fig. 5
Comparison of bca confidence limits for neonate coefficient resp, logistic regression MLE (solid) and glmnet (dashed).
Fig. 6
Fig. 6
Left panel: B = 16000 bootstrap replications of θ^* (4.27). Right panel: confidence limits for θ having observed θ^=1; green standard, black bca, red dashed exact. The corrections to the standard limits are enormous in this case.
Fig. 7
Fig. 7
Schematic diagram concerning estimation of the acceleration a, and its relation to the bias-corrector z0, as explained in the text.
See this image and copyright information in PMC

References

    1. Barndorff-Nielsen O (1983). On a formula for the distribution of the maximum likelihood estimator. Biometrika 70: 343–365, doi: 10.1093/biomet/70.2.343. - DOI
    1. Barndorff-Nielsen OE and Cox DR (1994). Inference and Asymptotics, Monographs on Statistics and Applied Probability 52. Chapman & Hall, London, doi: 10.1007/978-1-4899-3210-5. - DOI
    1. DiCiccio T and Efron B (1992). More accurate confidence intervals in exponential families. Biometrika 79: 231–245.
    1. DiCiccio T and Efron B (1996). Bootstrap confidence intervals. Statist. Sci 11: 189–228.
    1. DiCiccio TJ and Young GA (2008). Conditional properties of unconditional parametric bootstrap procedures for inference in exponential families. Biometrika 95: 747–758, doi: 10.1093/biomet/asn011. - DOI

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