## The Annals of Statistics

### Convolved subsampling estimation with applications to block bootstrap

#### Abstract

The block bootstrap approximates sampling distributions from dependent data by resampling data blocks. A fundamental problem is establishing its consistency for the distribution of a sample mean, as a prototypical statistic. We use a structural relationship with subsampling to characterize the bootstrap in a new and general manner. While subsampling and block bootstrap differ, the block bootstrap distribution of a sample mean equals that of a $k$-fold self-convolution of a subsampling distribution. Motivated by this, we provide simple necessary and sufficient conditions for a convolved subsampling estimator to produce a normal limit that matches the target of bootstrap estimation. These conditions may be linked to consistency properties of an original subsampling distribution, which are often obtainable under minimal assumptions. Through several examples, the results are shown to validate the block bootstrap for means under significantly weakened assumptions in many existing (and some new) dependence settings, which also addresses a standing conjecture of Politis, Romano and Wolf [Subsampling (1999) Springer]. Beyond sample means, convolved subsampling may not match the block bootstrap, but instead provides an alternative resampling estimator that may be of interest. Under minimal dependence conditions, results also broadly establish convolved subsampling for general statistics having normal limits.

#### Article information

Source
Ann. Statist., Volume 47, Number 1 (2019), 468-496.

Dates
Revised: February 2018
First available in Project Euclid: 30 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.aos/1543568595

Digital Object Identifier
doi:10.1214/18-AOS1695

Mathematical Reviews number (MathSciNet)
MR3909939

Zentralblatt MATH identifier
07036208

#### Citation

Tewes, Johannes; Politis, Dimitris N.; Nordman, Daniel J. Convolved subsampling estimation with applications to block bootstrap. Ann. Statist. 47 (2019), no. 1, 468--496. doi:10.1214/18-AOS1695. https://projecteuclid.org/euclid.aos/1543568595

#### References

• [1] Athreya, K. B. and Lahiri, S. N. (2006). Measure Theory and Probability Theory. Springer Texts in Statistics. Springer, New York.
• [2] Bai, S. and Taqqu, M. S. (2017). On the validity of resampling methods under long memory. Ann. Statist. 45 2365–2399.
• [3] Beran, J. (1994). Statistics for Long-Memory Processes. Monographs on Statistics and Applied Probability 61. Chapman & Hall, New York.
• [4] Betken, A. and Wendler, M. (2018). Subsampling for general statistics under long range dependence. Statist. Sinica. 28 1199–1224.
• [5] Bickel, P. J. and Freedman, D. A. (1981). Some asymptotic theory for the bootstrap. Ann. Statist. 9 1196–1217.
• [6] Cambanis, S., Houdré, C., Hurd, H. and Leśkow, J. (1994). Laws of large numbers for periodically and almost periodically correlated processes. Stochastic Process. Appl. 53 37–54.
• [7] Corduneanu, C. (1989). Almost Periodic Functions. Chelsea, New York.
• [8] Davydov, Y. A. (1970). The invariance principle for stationary processes. Theory Probab. Appl. 15 487-498.
• [9] Dehling, H. and Wendler, M. (2010). Central limit theorem and the bootstrap for $U$-statistics of strongly mixing data. J. Multivariate Anal. 101 126–137.
• [10] Dobrushin, R. L. and Major, P. (1979). Non-central limit theorems for nonlinear functionals of Gaussian fields. Z. Wahrsch. Verw. Gebiete 50 27–52.
• [11] Efron, B. (1979). Bootstrap methods: Another look at the jackknife. Ann. Statist. 7 1–26.
• [12] Fitzenberger, B. (1998). The moving blocks bootstrap and robust inference for linear least squares and quantile regressions. J. Econometrics 82 235–287.
• [13] Hosking, J. R. M. (1981). Fractional differencing. Biometrika 68 165–176.
• [14] Hurd, H. L. (1991). Correlation theory of almost periodically correlated processes. J. Multivariate Anal. 37 24–45.
• [15] Tewes, J., Politis, D. N. and Nordman, D. J. (2019). Supplement to “Convolved subsampling estimation with applications to block bootstrap.” DOI:10.1214/18-AOS1695SUPP.
• [16] Kim, Y. M. and Nordman, D. J. (2011). Properties of a block bootstrap under long-range dependence. Sankhya A 73 79–109.
• [17] Künsch, H. R. (1989). The jackknife and the bootstrap for general stationary observations. Ann. Statist. 17 1217–1241.
• [18] Lahiri, S. N. (1993). On the moving block bootstrap under long range dependence. Statist. Probab. Lett. 18 405–413.
• [19] Lahiri, S. N. (2003). Resampling Methods for Dependent Data. Springer Series in Statistics. Springer, New York.
• [20] Lahiri, S. N. (2003). Central limit theorems for weighted sums of a spatial process under a class of stochastic and fixed designs. Sankhya 65 356–388.
• [21] Lenart, Ł. (2011). Asymptotic distributions and subsampling in spectral analysis for almost periodically correlated time series. Bernoulli 17 290–319.
• [22] Lenart, Ł. (2016). Generalized resampling scheme with application to spectral density matrix in almost periodically correlated class of times series. J. Time Series Anal. 37 369–404.
• [23] Leucht, A. (2012). Degenerate $U$- and $V$-statistics under weak dependence: Asymptotic theory and bootstrap consistency. Bernoulli 18 552–585.
• [24] Liu, R. Y. (1988). Bootstrap procedures under some non-i.i.d. models. Ann. Statist. 16 1696–1708.
• [25] Liu, R. Y. and Singh, K. (1992). Moving blocks jackknife and bootstrap capture weak dependence. In Exploring the Limits of Bootstrap (East Lansing, MI, 1990). Wiley Ser. Probab. Math. Statist. Probab. Math. Statist. 225–248. Wiley, New York.
• [26] Mandelbrot, B. B. and Van Ness, J. W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 422–437.
• [27] Nordman, D. J. and Lahiri, S. N. (2005). Validity of the sampling window method for long-range dependent linear processes. Econometric Theory 21 1087–1111.
• [28] Politis, D. N. and Romano, J. P. (1994). Large sample confidence regions based on subsamples under minimal assumptions. Ann. Statist. 22 2031–2050.
• [29] Politis, D. N., Romano, J. P. and Wolf, M. (1999). Subsampling. Springer Series in Statistics. Springer, New York.
• [30] Radulović, D. (1996). The bootstrap of the mean for strong mixing sequences under minimal conditions. Statist. Probab. Lett. 28 65–72.
• [31] Radulović, D. (2012). Necessary and sufficient conditions for the moving blocks bootstrap central limit theorem of the mean. J. Nonparametr. Stat. 24 343–357.
• [32] Sharipov, O. Sh., Tewes, J. and Wendler, M. (2016). Bootstrap for $U$-statistics: A new approach. J. Nonparametr. Stat. 28 576–594.
• [33] Sharipov, O. Sh. and Wendler, M. (2012). Bootstrap for the sample mean and for $U$-statistics of mixing and near-epoch dependent processes. J. Nonparametr. Stat. 24 317–342.
• [34] Synowiecki, R. (2007). Consistency and application of moving block bootstrap for non-stationary time series with periodic and almost periodic structure. Bernoulli 13 1151–1178.
• [35] Taqqu, M. S. (1974/75). Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrsch. Verw. Gebiete 31 287–302.
• [36] Zhang, T., Ho, H.-C., Wendler, M. and Wu, W. B. (2013). Block sampling under strong dependence. Stochastic Process. Appl. 123 2323–2339.

#### Supplemental materials

• Supplement to “Convolved subsampling estimation with applications to block bootstrap”. This supplement provides proofs for the distributional results about convolved subsampling and further numerical/theoretical support for the simulation study.