A large deviation principle for bootstrapped sample means

Li, Deli; Rosalsky, Andrew; Al-Mutairi, Dhaifalla

doi:10.1090/S0002-9939-01-06368-7

A large deviation principle for bootstrapped sample means
HTML articles powered by AMS MathViewer

by Deli Li, Andrew Rosalsky and Dhaifalla K. Al-Mutairi PDF

Proc. Amer. Math. Soc. 130 (2002), 2133-2138 Request permission

Abstract:

A large deviation principle for bootstrapped sample means is established. It relies on the Bolthausen large deviation principle for sums of i.i.d. Banach space valued random variables. The rate function of the large deviation principle for bootstrapped sample means is the same as the classical one.

References

S.E. Ahmed, D. Li, A. Rosalsky, and A.I. Volodin, Almost sure lim sup behavior of bootstrapped means with applications to pairwise i.i.d. sequences and stationary ergodic sequences, J. Statist. Plann. Inference 98 (2001), 1-14.
Eusebio Arenal-Gutiérrez, Carlos Matrán, and Juan A. Cuesta-Albertos, Unconditional Glivenko-Cantelli-type theorems and weak laws of large numbers for bootstrap, Statist. Probab. Lett. 26 (1996), no. 4, 365–375. MR 1393922, DOI 10.1016/0167-7152(95)00033-X
Eusebio Arenal-Gutiérrez, Carlos Matrán, and Juan A. Cuesta-Albertos, On the unconditional strong law of large numbers for the bootstrap mean, Statist. Probab. Lett. 27 (1996), no. 1, 49–60. MR 1394176, DOI 10.1016/0167-7152(95)00043-7
K. B. Athreya, Strong law for the bootstrap, Statist. Probab. Lett. 1 (1983), no. 3, 147–150. MR 699258, DOI 10.1016/0167-7152(83)90063-9
Krishna B. Athreya, Malay Ghosh, Leone Y. Low, and Pranab K. Sen, Laws of large numbers for bootstrapped $U$-statistics, J. Statist. Plann. Inference 9 (1984), no. 2, 185–194. MR 738368, DOI 10.1016/0378-3758(84)90019-3
R. R. Bahadur and S. L. Zabell, Large deviations of the sample mean in general vector spaces, Ann. Probab. 7 (1979), no. 4, 587–621. MR 537209
Peter J. Bickel and David A. Freedman, Some asymptotic theory for the bootstrap, Ann. Statist. 9 (1981), no. 6, 1196–1217. MR 630103
Patrick Billingsley, Probability and measure, 2nd ed., Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. MR 830424
E. Bolthausen, On the probability of large deviations in Banach spaces, Ann. Probab. 12 (1984), no. 2, 427–435. MR 735846
Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
S. Chevet, Gaussian measures and large deviations, Unpublished manuscript (1982).
H. Cramér, Sur un nouveau théorème-limite de la théorie des probabilités, Actualités Sci. Indust. 736 (1938), 5-23.
Sándor Csörgő, On the law of large numbers for the bootstrap mean, Statist. Probab. Lett. 14 (1992), no. 1, 1–7. MR 1172283, DOI 10.1016/0167-7152(92)90203-H
S. Csörgő and W.B. Wu, Random graphs and the strong convegence of bootstrap means, Combin. Probab. Comput. 9 (2000), 315-347.
M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time. III, Comm. Pure Appl. Math. 29 (1976), no. 4, 389–461. MR 428471, DOI 10.1002/cpa.3160290405
B. Efron, Bootstrap methods: another look at the jackknife, Ann. Statist. 7 (1979), no. 1, 1–26. MR 515681
Franz Rádl, Über die Teilbarkeitsbedingungen bei den gewöhnlichen Differential polynomen, Math. Z. 45 (1939), 429–446 (German). MR 82, DOI 10.1007/BF01580293
Paul Erdős and Alfréd Rényi, On a new law of large numbers, J. Analyse Math. 23 (1970), 103–111. MR 272026, DOI 10.1007/BF02795493
Evarist Giné and Joel Zinn, Necessary conditions for the bootstrap of the mean, Ann. Statist. 17 (1989), no. 2, 684–691. MR 994259, DOI 10.1214/aos/1176347134
Peter Hall, On the relative performance of bootstrap and Edgeworth approximations of a distribution function, J. Multivariate Anal. 35 (1990), no. 1, 108–129. MR 1084945, DOI 10.1016/0047-259X(90)90019-E
Deli Li, Andrew Rosalsky, and S. E. Ahmed, Complete convergence of bootstrapped means and moments of the supremum of normed bootstrapped sums, Stochastic Anal. Appl. 17 (1999), no. 5, 799–814. MR 1714899, DOI 10.1080/07362999908809635
T. Mikosch, Almost sure convergence of bootstrapped means and $U$-statistics, J. Statist. Plann. Inference 41 (1994), no. 1, 1–19. MR 1292144, DOI 10.1016/0378-3758(94)90151-1
L. A. Shepp, A limit law concerning moving averages, Ann. Math. Statist. 35 (1964), 424–428. MR 166818, DOI 10.1214/aoms/1177703767
Kesar Singh, On the asymptotic accuracy of Efron’s bootstrap, Ann. Statist. 9 (1981), no. 6, 1187–1195. MR 630102