Moderate deviations in subsampling distribution estimation

Authors:
Patrice Bertail, Anthony Gamst and Dimitris N. Politis

Journal:
Proc. Amer. Math. Soc. **129** (2001), 551-557

MSC (1991):
Primary 60F05; Secondary 60F10

Published electronically:
July 27, 2000

MathSciNet review:
1707507

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

In Politis and Romano (1994) the subsampling methodology was put forth for approximating the sampling distribution (and the corresponding quantiles) of general statistics from i.i.d. and stationary data. In this note, we address the question of how well the subsampling distribution approximates the tail of the target distribution. In the regular setting of the sample mean of an -dependent sequence we show a moderate deviation property of the subsampling distribution.

**1.**Andrews, D. and Pollard, D. (1994), An introduction to functional central limit theorems for dependent stochastic processes,*Int. Stat. Rev.*, 62, 119-132.**2.**Lothar Heinrich,*A method for the derivation of limit theorems for sums of 𝑚-dependent random variables*, Z. Wahrsch. Verw. Gebiete**60**(1982), no. 4, 501–515. MR**665742**, 10.1007/BF00535713**3.**Lothar Heinrich,*Nonuniform estimates and asymptotic expansions of the remainder in the central limit theorem for 𝑚-dependent random variables*, Math. Nachr.**115**(1984), 7–20. MR**755264**, 10.1002/mana.19841150102**4.**Bing-Yi Jing,*On the relative performance of the block bootstrap for dependent data*, Comm. Statist. Theory Methods**26**(1997), no. 6, 1313–1328. MR**1456833**, 10.1080/03610929708831984**5.**Regina Y. Liu and Kesar Singh,*Moving blocks jackknife and bootstrap capture weak dependence*, Exploring the limits of bootstrap (East Lansing, MI, 1990) Wiley Ser. Probab. Math. Statist. Probab. Math. Statist., Wiley, New York, 1992, pp. 225–248. MR**1197787****6.**Dimitris N. Politis and Joseph P. Romano,*Large sample confidence regions based on subsamples under minimal assumptions*, Ann. Statist.**22**(1994), no. 4, 2031–2050. MR**1329181**, 10.1214/aos/1176325770**7.**David Pollard,*Convergence of stochastic processes*, Springer Series in Statistics, Springer-Verlag, New York, 1984. MR**762984****8.**David Pollard,*Bracketing methods in statistics and econometrics*, Nonparametric and semiparametric methods in econometrics and statistics (Durham, NC, 1988) Internat. Sympos. Econom. Theory Econometrics, Cambridge Univ. Press, Cambridge, 1991, pp. 337–355. MR**1174979**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
60F05,
60F10

Retrieve articles in all journals with MSC (1991): 60F05, 60F10

Additional Information

**Patrice Bertail**

Affiliation:
INRA-CORELA, 65, Bd. de Brandebourg, 34205 Ivry-Seine, France

**Anthony Gamst**

Affiliation:
Department of Mathematics, University of California at San Diego, La Jolla, California 92093-0112

Email:
acgamst@osiris.ucsd.edu

**Dimitris N. Politis**

Affiliation:
Department of Mathematics, University of California at San Diego, La Jolla, California 92093-0112

Email:
politis@euclid.ucsd.edu

DOI:
https://doi.org/10.1090/S0002-9939-00-05551-9

Keywords:
Central limit theorem,
large deviations,
resampling

Received by editor(s):
April 6, 1998

Received by editor(s) in revised form:
April 30, 1999

Published electronically:
July 27, 2000

Communicated by:
Stanley Sawyer

Article copyright:
© Copyright 2000
American Mathematical Society