Electronic Journal of Statistics

Bootstrap consistency for quadratic forms of sample averages with increasing dimension

Demian Pouzo

Full-text: Open access

Abstract

This paper establishes consistency of the weighted bootstrap for quadratic forms $\left(n^{-1/2}\sum_{i=1}^{n}Z_{i,n} \right)^{T}\left(n^{-1/2}\sum_{i=1}^{n}Z_{i,n}\right)$ where $(Z_{i,n})_{i=1}^{n}$ are mean zero, independent $\mathbb{R}^{d}$-valued random variables and $d=d(n)$ is allowed to grow with the sample size $n$, slower than $n^{1/4}$. The proof relies on an adaptation of Lindeberg interpolation technique whereby we simplify the original problem to a Gaussian approximation problem. We apply our bootstrap results to model-specification testing problems when the number of moments is allowed to grow with the sample size.

Article information

Source
Electron. J. Statist., Volume 9, Number 2 (2015), 3046-3097.

Dates
Received: December 2014
First available in Project Euclid: 19 January 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1453212364

Digital Object Identifier
doi:10.1214/15-EJS1090

Mathematical Reviews number (MathSciNet)
MR3450756

Zentralblatt MATH identifier
1384.62157

Subjects
Primary: 60F05: Central limit and other weak theorems 60F17: Functional limit theorems; invariance principles 62E20: Asymptotic distribution theory 62F40: Bootstrap, jackknife and other resampling methods

Keywords
Asymptotic theory bootstrap high dimensional statistics quadratic forms model specification test

Citation

Pouzo, Demian. Bootstrap consistency for quadratic forms of sample averages with increasing dimension. Electron. J. Statist. 9 (2015), no. 2, 3046--3097. doi:10.1214/15-EJS1090. https://projecteuclid.org/euclid.ejs/1453212364


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References

  • Arellano, M. and Bond, S. Some tests of specification for panel data: Monte carlo evidence and an application to employment equations., The Review of Economic Studies, 58(2):277–297, 1991.
  • Boucheron, S., Lugosi, G., and Massart, P., Concentration Inequalities. Oxford Univ. Press, 2013.
  • Chatterjee, S. A generalization of the Lindeberg principle., The Annals of Probability, 34(6) :2061–2076, 2006.
  • Chatterjee, S. and Meckes, E. Multivariate normal approximation using exchangeable pairs., ALEA Lat. Am. J. Probab. Math. Stat., 4:257–283, 2008.
  • Chen, X. and Pouzo, D. Sieve Wald and QLR inferences on semi/nonparametric conditional moment models., Econometrica, 83(3) :1013–1079, May 2015.
  • Chen, X. Large sample sieve estimation of semi-nonparametric models. Part B of, Handbook of Econometrics, Vol. 6, chapter 76, pages 5549–5632. Elsevier, 2007.
  • Chernozhukov, V., Chetverikov, D., and Kato, K. Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors., The Annals of Statistics, 41 :2786–2819, 2013a.
  • Chernozhukov, V., Chetverikov, D., and Kato, K. Comparison and anti-concentration bounds for maxima of Gaussian random vectors., arXiv:1301.4807, 2013b.
  • de Jong, R. M. and Bierens, H. J. On the limit behavior of a Chi-Square type test if the number of conditional moments tested approaches infinity., Econometric Theory, 10(01):70–90, 1994.
  • Donald, S., Imbens, G., and Newey, W. Empirical likelihood estimation and consistent tests with conditional moment restrictions., Journal of Econometrics, 117:55–93, 2003.
  • Efron, B. Bootstrap methods: Another look at the jackknife., The Annals of Statistics, 7(1):1–26, 1979.
  • Feller, W., An Introduction to Probability Theory and its Applications, Vol. II. Wiley, 2nd edition, 1971.
  • Hall, A., Generalized Method of Moments. Advanced Texts in Econometrics Series. Oxford University Press, 2005.
  • Hall, P. Methodology and theory for the bootstrap. In Engle, R. F. and McFadden, D., editors, Handbook of Econometrics, Vol. 4, chapter 39, pages 2341–2381. Elsevier, 1 edition, 1986.
  • Hansen, L. P. Large sample properties of generalized method of moments estimators., Econometrica, 50(4) :1029–1054, 1982.
  • Hansen, L. P., Heaton, J., and Yaron, A. Finite-sample properties of some alternative GMM estimators., Journal of Business and Economic Statistics, 14(3):262–280, 1996.
  • He, X. and Shao, Q.-M. On parameters of increasing dimensions., Journal of Multivariate Analysis, 73:120–135, 2000.
  • Hjort, N., McKeague, I., and Van Keilegom, I. Extending the scope of empirical likelihood., The Annals of Statistics, 37(3) :1079–1111, 2009.
  • Horowitz, J. L. The Bootstrap. In Engle, R. F. and McFadden, D., editors, Handbook of Econometrics, Vol. 5, chapter 52, pages 3159–3228. Elsevier, 1 edition, 2001.
  • Imbens, G. W. Generalized method of moments and empirical likelihood., Journal of Business and Economic Statistics, 20(4):493–506, 2002.
  • Imbens, G. W., Spady, R. H., and Johnson, P. Information-theoretic approaches to inference in moment condition models., Econometrica, 66(2):333–358, 1998.
  • Johnson, W. B., Schechtman, G., and Zinn, J. Best constants in moment inequalities for linear combinations of independent and exchangeable random variables., The Annals of Probability, 13(1):234–253, 1985.
  • Kitamura, Y. and Stutzer, M. An information-theoretic alternative to generalized method of moments estimation., Econometrica, 65(4):861–874, 1997.
  • Koenker, R. and Machado, J. A. F. GMM inference when the number of moment conditions is large., Journal of Econometrics, 93(2):327–344, 1999.
  • Ma, S. and Kosorok, M. R. Robust semiparametric m-estimation and the weighted bootstrap., Journal of Multivariate Analysis, 96(1):190–217, 2005.
  • Mammen, E. Asymptotics with increasing dimension for robust regression with applications to the bootstrap., The Annals of Statistics, 17(61):382–400, 1989.
  • Mammen, E. Bootstrap and Wild bootstrap for high dimensional linear models., The Annals of Statistics, 21(1):255–285, 1993.
  • Murphy, S. and Van der Vaart, A. On profile likelihood. 95:449–485, 2000.
  • Newey, W. K. and McFadden, D. arge sample estimation and hypothesis testing., Handbook of Econometrics, Vol. 4, chapter 36, pages 2111–2245. Elsevier, 1994.
  • Owen, A., Empirical Likelihood. Chapman and Hall/CRC, 1990.
  • Owen, A. Empirical likelihood ratio confidence intervals for a single functional., Biometrika, 75(2):237–249, 1988.
  • Peng, H. and Schick, A. Asymptotic normality of quadratic forms with random vectors of increasing dimension., Working Paper, 2012.
  • Pollard, D., A User’s Guide to Measure Theoretic Probability. Cambridge University Press, 2001.
  • Portnoy, S. Asymptotic behavior of likelihood methods for exponential families when the number of parameters tends to infinity., The Annals of Statistics, 16(1):356–366, 1988.
  • Praestgaard, J. Bootstrap with general weights and multiplier central limit theorems., Technical Report 195, Dept. Statistics, Univ. Washington., 1990.
  • Praestgaard, J. and Wellner, J. A. Exchangeably weighted bootstraps of the general empirical process., The Annals of Probability, 21(4) :2053–2086, 1993.
  • Radulovic, D. Can we bootstrap even if CLT fails?, Journal of Theoretical Probability, 11(3):813–830, 1998.
  • Raic, M. A multivariate CLT for decomposable random vectors with finite second moments., Journal of Theoretical Probability, 17(3):573–603, 2004.
  • Rollin, A. Stein’s method in high dimensions with applications., arXiv:1101.4454, 2013.
  • Slepian, D. The one-sided barrier problem for Gaussian noise., Bell System Technical Journal, 41(2):463–501, 1962.
  • Smith, R. J. Alternative semi-parametric likelihood approaches to generalized method of moments estimation., The Economic Journal, 107(441):503–519, 1997.
  • Spokoiny, V. and Zhilova, M. Bootstrap confidence sets under a model misspecification., arXiv:1410.0347v1, 2014.
  • Stein, C. Estimation of the mean of a multivariate normal distribution., The Annals of Statistics, 9(6) :1135–1151, 1981.
  • Tao, T. and Vu, V. Random matrices: The Four Moment Theorem for Wigner ensembles., arXiv:1112.1976, 2011.
  • Van der Vaart, A. and Wellner, J., Weak Convergence and Empirical Processes. Springer, 1996.
  • Van der Vaart, A., Asymptotic Statistics. Cambridge University Press, 2000.
  • Vershynin, R. Introduction to the non-asymptotic analysis of random matrices. In, Compressed sensing, pages 210–268. Cambridge Univ. Press, 2012a.
  • Vershynin, R. How close is the sample covariance matrix to the actual covariance matrix?, Journal of Theoretical Probability, 25:655–686, 2012b.
  • Wasserman, L. Stein’s method and the bootstrap in low and high dimensions: A tutorial., Working Paper, 2014.
  • Xu, M., Zhang, D., and Wu, W. B. $L^2$ asymptotics for high-dimensional data., arXiv:1405.7244, 2014.
  • Zhang, X. and Cheng, G. Bootstrapping high dimensional time series., arXiv:1406.1037v2, 2014.