## The Annals of Statistics

### Bootstrap confidence sets under model misspecification

#### Abstract

A multiplier bootstrap procedure for construction of likelihood-based confidence sets is considered for finite samples and a possible model misspecification. Theoretical results justify the bootstrap validity for a small or moderate sample size and allow to control the impact of the parameter dimension $p$: the bootstrap approximation works if $p^{3}/n$ is small. The main result about bootstrap validity continues to apply even if the underlying parametric model is misspecified under the so-called small modelling bias condition. In the case when the true model deviates significantly from the considered parametric family, the bootstrap procedure is still applicable but it becomes a bit conservative: the size of the constructed confidence sets is increased by the modelling bias. We illustrate the results with numerical examples for misspecified linear and logistic regressions.

#### Article information

Source
Ann. Statist., Volume 43, Number 6 (2015), 2653-2675.

Dates
Revised: June 2015
First available in Project Euclid: 7 October 2015

https://projecteuclid.org/euclid.aos/1444222088

Digital Object Identifier
doi:10.1214/15-AOS1355

Mathematical Reviews number (MathSciNet)
MR3405607

Zentralblatt MATH identifier
1327.62179

#### Citation

Spokoiny, Vladimir; Zhilova, Mayya. Bootstrap confidence sets under model misspecification. Ann. Statist. 43 (2015), no. 6, 2653--2675. doi:10.1214/15-AOS1355. https://projecteuclid.org/euclid.aos/1444222088

#### References

• Aerts, M. and Claeskens, G. (2001). Bootstrap tests for misspecified models, with application to clustered binary data. Comput. Statist. Data Anal. 36 383–401.
• Arlot, S., Blanchard, G. and Roquain, E. (2010). Some nonasymptotic results on resampling in high dimension. I. Confidence regions. Ann. Statist. 38 51–82.
• Barbe, P. and Bertail, P. (1995). The Weighted Bootstrap. Lecture Notes in Statistics 98. Springer, New York.
• Bücher, A. and Dette, H. (2013). Multiplier bootstrap of tail copulas with applications. Bernoulli 19 1655–1687.
• Chatterjee, S. and Bose, A. (2005). Generalized bootstrap for estimating equations. Ann. Statist. 33 414–436.
• Chen, X. and Pouzo, D. (2009). Efficient estimation of semiparametric conditional moment models with possibly nonsmooth residuals. J. Econometrics 152 46–60.
• Chen, X. and Pouzo, D. (2015). Sieve Wald and QLR inferences on semi/nonparametric conditional moment models. Econometrica 83 1013–1079.
• Chernozhukov, V., Chetverikov, D. and Kato, K. (2013). Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors. Ann. Statist. 41 2786–2819.
• Efron, B. (1979). Bootstrap methods: Another look at the jackknife. Ann. Statist. 7 1–26.
• Hall, P. (1992). The Bootstrap and Edgeworth Expansion. Springer, New York.
• Hall, A. R. (2005). Generalized Method of Moments. Oxford Univ. Press, Oxford.
• Horowitz, J. L. (2001). The bootstrap. Handbook of Econometrics 5 3159–3228.
• Huber, P. J. (1967). The behavior of maximum likelihood estimates under nonstandard conditions. In Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), Vol. I: Statistics 221–233. Univ. California Press, Berkeley, CA.
• Janssen, P. (1994). Weighted bootstrapping of $U$-statistics. J. Statist. Plann. Inference 38 31–41.
• Janssen, A. and Pauls, T. (2003). How do bootstrap and permutation tests work? Ann. Statist. 31 768–806.
• Kline, P. and Santos, A. (2012). Higher order properties of the wild bootstrap under misspecification. J. Econometrics 171 54–70.
• Koenker, R. and Bassett, G. Jr. (1978). Regression quantiles. Econometrica 46 33–50.
• Lavergne, P. and Patilea, V. (2013). Smooth minimum distance estimation and testing with conditional estimating equations: Uniform in bandwidth theory. J. Econometrics 177 47–59.
• Liu, R. Y. (1988). Bootstrap procedures under some non-i.i.d. models. Ann. Statist. 16 1696–1708.
• Ma, S. and Kosorok, M. R. (2005). Robust semiparametric M-estimation and the weighted bootstrap. J. Multivariate Anal. 96 190–217.
• Mammen, E. (1992). When Does Bootstrap Work? Lecture Notes in Statistics 77. Springer, New York.
• Mammen, E. (1993). Bootstrap and wild bootstrap for high-dimensional linear models. Ann. Statist. 21 255–285.
• Newton, M. A. and Raftery, A. E. (1994). Approximate Bayesian inference with the weighted likelihood bootstrap. J. R. Stat. Soc. Ser. B. Stat. Methodol. 56 3–48.
• Spokoiny, V. (2012). Parametric estimation. Finite sample theory. Ann. Statist. 40 2877–2909.
• Spokoiny, V. (2013). Bernstein–von Mises theorem for growing parameter dimension. Preprint. Available at arXiv:1302.3430.
• Spokoiny, V. and Zhilova, M. (2015). Supplement to “Bootstrap confidence sets under model misspecification.” DOI:10.1214/15-AOS1355SUPP.
• van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer, New York.
• Wilks, S. S. (1938). The large-sample distribution of the likelihood ratio for testing composite hypotheses. The Annals of Mathematical Statistics 9 60–62.
• Wu, C.-F. J. (1986). Jackknife, bootstrap and other resampling methods in regression analysis. Ann. Statist. 14 1261–1350.

#### Supplemental materials

• Supplement to “Bootstrap confidence sets under model misspecification”. The supplementary material contains a proof of the square-root Wilks approximation for the bootstrap world, proofs of the main results from Section 2, and results on Gaussian approximation for $\ell_{2}$-norm of a sum of independent vectors.