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Consistency of the jackknife-after-bootstrap variance estimator for the bootstrap quantiles of a Studentized statistic. (English) Zbl 1086.62033
Summary: B. Efron [J. R. Stat. Soc., Ser. B 54, No. 1, 83–127 (1992; Zbl 0782.62051]] proposed a computationally efficient method, called the jackknife-after-bootstrap, for estimating the variance of a bootstrap estimator for independent data. For dependent data, a version of the jackknife-after-bootstrap method has been recently proposed by S. N. Lahiri [Econom. Theory 18, No. 1, 79–98 (2002; Zbl 1181.62058]). In this paper it is shown that the jackknife-after-bootstrap estimators of the variance of a bootstrap quantile are consistent for both dependent and independent data. Results from a simulation study are also presented.

##### MSC:
 62F12 Asymptotic properties of parametric estimators 62F40 Bootstrap, jackknife and other resampling methods 62G09 Nonparametric statistical resampling methods 62G05 Nonparametric estimation 65C60 Computational problems in statistics (MSC2010) 62G20 Asymptotic properties of nonparametric inference
##### Keywords:
block bootstrap; consistency; weak dependence; tables
Full Text:
##### References:
 [1] Bhattacharya, R. N. and Ghosh, J. K. (1978). On the validity of the formal Edgeworth expansion. Ann. Statist. 6 434–451. · Zbl 0396.62010 [2] Bhattacharya, R. N. and Qumsiyeh, M. (1989). Second order and $$L^p$$-comparisons between the bootstrap and empirical Edgeworth expansion methodologies. Ann. Statist. 17 160–169. · Zbl 0669.62002 [3] Bhattacharya, R. N. and Ranga Rao, R. (1986). Normal Approximation and Asymptotic Expansions . Krieger, Malabar, FL. · Zbl 0657.41001 [4] Bose, A. (1988). Edgeworth correction by bootstrap in autoregressions. Ann. Statist. 16 1709–1722. · Zbl 0653.62016 [5] Bühlmann, P. (1994). Blockwise bootstrapped empirical process for stationary sequences. Ann. Statist. 22 995–1012. · Zbl 0806.62032 [6] Bustos, O. (1982). General $$M$$-estimates for contaminated $$p$$-th order autoregressive processes: Consistency and asymptotic normality. Z. Wahrsch. Verw. Gebiete 59 491–504. · Zbl 0482.62080 [7] Carlstein, E. (1986). The use of subseries values for estimating the variance of a general statistic from a stationary sequence. Ann. Statist. 14 1171–1179. · Zbl 0602.62029 [8] Datta, S. and McCormick, W. P. (1995). Bootstrap inference for a first-order autoregression with positive innovations. J. Amer. Statist. Assoc. 90 1289–1300. · Zbl 0868.62068 [9] Efron, B. (1979). Bootstrap methods: Another look at the jackknife. Ann. Statist. 7 1–26. · Zbl 0406.62024 [10] Efron, B. (1982). The Jackknife , the Bootstrap and Other Resampling Plans. SIAM, Philadelphia. · Zbl 0496.62036 [11] Efron, B. (1992). Jackknife-after-bootstrap standard errors and influence functions (with discussion). J. Roy. Statist. Soc. Ser. B 54 83–127. · Zbl 0782.62051 [12] Fuk, D. H. and Nagaev, S. V. (1971). Probability inequalities for sums of independent random variables. Theory Probab. Appl. 16 643–660. · Zbl 0259.60024 [13] Fukuchi, J.-I. (1994). Bootstrapping extremes of random variables. Ph.D. dissertation, Iowa State Univ., Ames, IA. [14] Götze, F. and Hipp, C. (1978). Asymptotic expansions in the central limit theorem under moment conditions. Z. Wahrsch. Verw. Gebiete 42 67–87. · Zbl 0369.60027 [15] Götze, F. and Hipp, C. (1983). Asymptotic expansions for sums of weakly dependent random vectors. Z. Wahrsch. Verw. Gebiete 64 211–239. · Zbl 0497.60022 [16] Götze, F. and Hipp, C. (1994). Asymptotic distribution of statistics in time series. Ann. Statist. 22 2062–2088. · Zbl 0827.62015 [17] Götze, F. and Künsch, H. R. (1996). Second-order correctness of the blockwise bootstrap for stationary observations. Ann. Statist. 24 1914–1933. · Zbl 0906.62040 [18] Hall, P. (1985). Resampling a coverage pattern. Stochastic Process. Appl. 20 231–246. · Zbl 0587.62081 [19] Hall, P. (1992). The Bootstrap and Edgeworth Expansion . Springer. New York. · Zbl 0744.62026 [20] Hall, P., Horowitz, J. L. and Jing, B.-Y. (1995). On blocking rules for the bootstrap with dependent data. Biometrika 82 561–574. · Zbl 0830.62082 [21] Hill, R. C., Cartwright, P. A. and Arbaugh, J. F. (1997). Jackknifing the bootstrap: Some Monte Carlo evidence. Comm. Statist. Simulation Comput. 26 125–139. · Zbl 0925.62072 [22] Künsch, H. R. (1989). The jackknife and the bootstrap for general stationary observations. Ann. Statist. 17 1217–1241. · Zbl 0684.62035 [23] Lahiri, S. N. (1991). Second order optimality of stationary bootstrap. Statist. Probab. Lett. 11 335–341. · Zbl 0722.62016 [24] Lahiri, S. N. (1993). Refinements in asymptotic expansions for sums of weakly dependent random vectors. Ann. Probab. 21 791–799. · Zbl 0776.60025 [25] Lahiri, S. N. (1996). On Edgeworth expansion and moving block bootstrap for Studentized $$M$$-estimators in multiple linear regression models. J. Multivariate Anal. 56 42–59. · Zbl 0864.62028 [26] Lahiri, S. N. (1999). Theoretical comparisons of block bootstrap methods. Ann. Statist. 27 386–404. · Zbl 0945.62049 [27] Lahiri, S. N. (2002). On the jackknife-after-bootstrap method for dependent data and its consistency properties. Econometric Theory 18 79–98. · Zbl 1181.62058 [28] Lahiri, S. N. (2003). Resampling Methods for Dependent Data . Springer, New York. · Zbl 1028.62002 [29] Lahiri, S. N., Furukawa, K. and Lee, Y.-D. (2003). A nonparametric plug-in rule for selecting optimal block lengths for block bootstrap methods. Preprint, Dept. Statistics, Iowa State Univ., Ames, IA. · Zbl 1248.62060 [30] Liu, R. Y. and Singh, K. (1992). Moving blocks jackknife and bootstrap capture weak dependence. In Exploring the Limits of Bootstrap (R. LePage and L. Billard, eds.) 225–248. Wiley, New York. · Zbl 0838.62036 [31] Miller, R. G. (1974). The jackknife—a review. Biometrika 61 1–15. · Zbl 0275.62035 [32] Naik-Nimbalkar, U. V. and Rajarshi, M. B. (1994). Validity of blockwise bootstrap for empirical processes with stationary observations. Ann. Statist. 22 980–994. · Zbl 0808.62043 [33] Paparoditis, E. and Politis, D. N. (2002). The tapered block bootstrap for general statistics from stationary sequences. Econom. J. 5 131–148. · Zbl 1009.62034 [34] Parr, W. C. and Schucany, W. R. (1982). Jackknifing $$L$$-statistics with smooth weight functions. J. Amer. Statist. Assoc. 77 629–638. · Zbl 0504.62042 [35] Politis, D. N. and Romano, J. P. (1992). A circular block-resampling procedure for stationary data. In Exploring the Limits of Bootstrap (R. LePage and L. Billard, eds.) 263–270. Wiley, New York. · Zbl 0845.62036 [36] Politis, D. N. and Romano, J. P. (1992). A general resampling scheme for triangular arrays of $$\alpha$$-mixing random variables with application to the problem of spectral density estimation. Ann. Statist. 20 1985–2007. · Zbl 0776.62070 [37] Politis, D. N. and Romano, J. P. (1994). Stationary bootstrap. J. Amer. Statist. Assoc. 89 1303–1313. · Zbl 0814.62023 [38] Shao, J. and Tu, D. S. (1995). The Jackknife and Bootstrap . Springer, New York. · Zbl 0947.62501 [39] Shao, J. and Wu, C. F. J. (1989). A general theory for jackknife variance estimation. Ann. Statist. 17 1176–1197. · Zbl 0684.62034
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