The Annals of Statistics

Bootstrap confidence sets under model misspecification

Vladimir Spokoiny and Mayya Zhilova

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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
Received: November 2014
Revised: June 2015
First available in Project Euclid: 7 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.aos/1444222088

Digital Object Identifier
doi:10.1214/15-AOS1355

Mathematical Reviews number (MathSciNet)
MR3405607

Zentralblatt MATH identifier
1327.62179

Subjects
Primary: 62F25: Tolerance and confidence regions 62F40: Bootstrap, jackknife and other resampling methods
Secondary: 62E17: Approximations to distributions (nonasymptotic)

Keywords
Likelihood-based bootstrap confidence set finite sample size multiplier/weighted bootstrap Gaussian approximation Pinsker’s inequality

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


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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.