A large deviation principle for bootstrapped sample means

Authors:
Deli Li, Andrew Rosalsky and Dhaifalla K. Al-Mutairi

Journal:
Proc. Amer. Math. Soc. **130** (2002), 2133-2138

MSC (1991):
Primary 60F10, 62G09; Secondary 60B12, 62G20

DOI:
https://doi.org/10.1090/S0002-9939-01-06368-7

Published electronically:
December 31, 2001

MathSciNet review:
1896050

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[1]**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.**[2]**E. Arenal-Gutiérrez, C. Matrán, and J.A. Cuesta-Albertos,*Unconditional Glivenko-Cantelli-type theorems and weak laws of large numbers for bootstrap,*Statist. Probab. Lett.**26**(1996), 365-375. MR**97c:60048****[3]**E. Arenal-Gutiérrez, C. Matrán, and J.A. Cuesta-Albertos,*On the unconditional strong law of large numbers for the bootstrap mean,*Statist. Probab. Lett.**27**(1996), 49-60. MR**97h:60020****[4]**K.B. Athreya,*Strong law for the bootstrap,*Statist. Probab. Lett.**1**(1983), 147-150. MR**84g:62026****[5]**K.B. Athreya, M. Ghosh, L.Y. Low, and P.K. Sen,*Laws of large numbers for bootstrapped -statistics,*J. Statist. Plann. Inference**9**(1984), 185-194. MR**85m:62062****[6]**R.R. Bahadur and S.L. Zabell,*Large deviations of the sample mean in general vector spaces,*Ann. Probab.**7**(1979), 587-621. MR**80i:60031****[7]**P.J. Bickel and D.A. Freedman,*Some asymptotic theory for the bootstrap,*Ann. Statist.**9**(1981), 1196-1217. MR**83a:62051****[8]**P. Billingsley,*Probability and Measure*, 2nd ed., Wiley, New York, 1986. MR**87f:60001****[9]**E. Bolthausen,*On the probability of large deviations in Banach spaces,*Ann. Probab.**12**(1984), 427-435. MR**85k:60011****[10]**H. Chernoff,*A measure of asymptotic efficiency for tests of hypothesis based on the sum of observations,*Ann. Math. Statist.**23**(1952), 493-507. MR**15:241c****[11]**S. Chevet,*Gaussian measures and large deviations,*Unpublished manuscript (1982).**[12]**H. Cramér,*Sur un nouveau théorème-limite de la théorie des probabilités,*Actualités Sci. Indust.**736**(1938), 5-23.**[13]**S. Csörgo,*On the law of large numbers for the bootstrap mean,*Statist. Probab. Lett.**14**(1992), 1-7. MR**93g:60063****[14]**S. Csörgo and W.B. Wu,*Random graphs and the strong convegence of bootstrap means,*Combin. Probab. Comput.**9**(2000), 315-347. CMP**2001:02****[15]**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), 389-461. MR**55:1492****[16]**B. Efron,*Bootstrap methods: Another look at the jackknife,*Ann. Statist.**7**(1979), 1-26. MR**80b:62021****[17]**R.S. Ellis and J.S. Rosen,*Laplace's method for Gaussian integrals with an application to statistical mechanics,*Ann. Probab.**10**(1982), 47-66. MR**82m:60010**; MR**84e:60014****[18]**P. Erdös and A. Rényi,*On a new law of large numbers,*J. Analyse Math.**23**(1970), 103-111. MR**42:6907****[19]**E. Giné and J. Zinn,*Necessary conditions for the bootstrap of the mean,*Ann. Statist.**17**(1989), 684-691. MR**90e:62030****[20]**P. Hall,*On the relative performance of bootstrap and Edgeworth approximations of a distribution function,*J. Multivariate Anal.**35**(1990), 108-129. MR**91m:62077****[21]**D. Li, A. Rosalsky, and S.E. Ahmed,*Complete convergence of bootstrapped means and moments of the supremum of normed boostrapped sums,*Stochastic Anal. Appl.**17**(1999), 799-814. MR**2001g:60067****[22]**T. Mikosch,*Amost sure convergence of bootstrapped means and -statistics,*J. Statist. Plann. Inference**41**(1994), 1-19. MR**95h:60046****[23]**L.A. Shepp,*A limit theorem concerning moving averages,*Ann. Math. Statist.**35**(1964), 424-428. MR**29:4091****[24]**K. Singh,*On the asymptotic accuracy of Efron's bootstrap,*Ann. Statist.**9**(1981), 1187-1195. MR**83c:62047**

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Additional Information

**Deli Li**

Affiliation:
Department of Mathematics & Statistics, Lakehead University, Thunder Bay, Ontario, Canada P7B 5E1

Email:
dli@sleet.lakeheadu.ca

**Andrew Rosalsky**

Affiliation:
Department of Statistics, University of Florida, P.O. Box 118545, Gainesville, Florida 32611

Email:
rosalsky@stat.ufl.edu

**Dhaifalla K. Al-Mutairi**

Affiliation:
Department of Statistics & Operations Research, Kuwait University, P.O. Box 21, Khaldiya 72461, Kuwait

Email:
dhaif@kuc01.kuniv.edu.kw

DOI:
https://doi.org/10.1090/S0002-9939-01-06368-7

Keywords:
Bootstrapped sample means,
large deviation principle,
Banach space valued random variables

Received by editor(s):
February 3, 2000

Received by editor(s) in revised form:
February 15, 2001

Published electronically:
December 31, 2001

Additional Notes:
The research of the first author was supported by a grant from the Natural Sciences and Engineering Research Council of Canada.

Communicated by:
Richard A. Davis

Article copyright:
© Copyright 2001
American Mathematical Society