## The Annals of Statistics

### On the asymptotic theory of new bootstrap confidence bounds

#### Abstract

We propose a new method, based on sample splitting, for constructing bootstrap confidence bounds for a parameter appearing in the regular smooth function model. It has been demonstrated in the literature, for example, by Hall [Ann. Statist. 16 (1988) 927–985; The Bootstrap and Edgeworth Expansion (1992) Springer], that the well-known percentile-$t$ method for constructing bootstrap confidence bounds typically incurs a coverage error of order $O(n^{-1})$, with $n$ being the sample size. Our version of the percentile-$t$ bound reduces this coverage error to order $O(n^{-3/2})$ and in some cases to $O(n^{-2})$. Furthermore, whereas the standard percentile bounds typically incur coverage error of $O(n^{-1/2})$, the new bounds have reduced error of $O(n^{-1})$. In the case where the parameter of interest is the population mean, we derive for each confidence bound the exact coefficient of the leading term in an asymptotic expansion of the coverage error, although similar results may be obtained for other parameters such as the variance, the correlation coefficient, and the ratio of two means. We show that equal-tailed confidence intervals with coverage error at most $O(n^{-2})$ may be obtained from the newly proposed bounds, as opposed to the typical error $O(n^{-1})$ of the standard intervals. It is also shown that the good properties of the new percentile-$t$ method carry over to regression problems. Results of independent interest are derived, such as a generalisation of a delta method by Cramér [Mathematical Methods of Statistics (1946) Princeton Univ. Press] and Hurt [Apl. Mat. 21 (1976) 444–456], and an expression for a polynomial appearing in an Edgeworth expansion of the distribution of a Studentised statistic for the slope parameter in a regression model. A small simulation study illustrates the behavior of the confidence bounds for small to moderate sample sizes.

#### Article information

Source
Ann. Statist., Volume 46, Number 1 (2018), 438-456.

Dates
Revised: January 2017
First available in Project Euclid: 22 February 2018

https://projecteuclid.org/euclid.aos/1519268436

Digital Object Identifier
doi:10.1214/17-AOS1557

Mathematical Reviews number (MathSciNet)
MR3766958

Zentralblatt MATH identifier
06865117

Subjects
Primary: 62G09: Resampling methods 62G20: Asymptotic properties
Secondary: 62G15: Tolerance and confidence regions

#### Citation

Pretorius, Charl; Swanepoel, Jan W. H. On the asymptotic theory of new bootstrap confidence bounds. Ann. Statist. 46 (2018), no. 1, 438--456. doi:10.1214/17-AOS1557. https://projecteuclid.org/euclid.aos/1519268436

#### References

• [1] Bhattacharya, R. N. and Ghosh, J. K. (1978). On the validity of the formal Edgeworth expansion. Ann. Statist. 6 434–451.
• [2] Bickel, P. J., Götze, F. and van Zwet, W. R. (1997). Resampling fewer than $n$ observations: Gains, losses, and remedies for losses. Statist. Sinica 7 1–31.
• [3] Chang, C. C. and Politis, D. N. (2011). Bootstrap with larger resample size for root-$n$ consistent density estimation with time series data. Statist. Probab. Lett. 81 652–661.
• [4] Chung, K.-H. and Lee, S. M. S. (2001). Optimal bootstrap sample size in construction of percentile confidence bounds. Scand. J. Stat. 28 225–239.
• [5] Cramér, H. (1946). Mathematical Methods of Statistics. Princeton Mathematical Series 9. Princeton Univ. Press, Princeton, NJ.
• [6] Efron, B. (1979). Bootstrap methods: Another look at the jackknife. Ann. Statist. 7 1–26.
• [7] Efron, B. and Tibshirani, R. J. (1993). An Introduction to the Bootstrap. Monographs on Statistics and Applied Probability 57. Chapman & Hall, New York.
• [8] Finner, H. and Dickhaus, T. (2010). Edgeworth expansions and rates of convergence for normalized sums: Chung’s 1946 method revisited. Statist. Probab. Lett. 80 1875–1880.
• [9] Hall, P. (1988). Theoretical comparison of bootstrap confidence intervals. Ann. Statist. 16 927–985.
• [10] Hall, P. (1992). The Bootstrap and Edgeworth Expansion. Springer, New York.
• [11] Hurt, J. (1976). Asymptotic expansions of functions of statistics. Apl. Mat. 21 444–456.
• [12] Pretorius, C. and Swanepoel, J. W. H. (2018). Supplement to “On the asymptotic theory of new bootstrap confidence bounds”. DOI:10.1214/17-AOS1557SUPP.
• [13] Swanepoel, J. W. H. (1986). A note on proving that the (modified) bootstrap works. Comm. Statist. Theory Methods 15 3193–3203.

#### Supplemental materials

• Supplement to “On the asymptotic theory of new bootstrap confidence bounds”. In the online supplement [12], we supply proofs for all theorems found in the main text.