Discussiones Mathematicae Probability and Statistics 21(2) (2001) 121-147


 Jan Ćwik

Institute of Computer Science, Polish Academy of Sciences
Ordona 21, 01-237 Warsaw, Poland
e-mail: jc@ipipan.waw.pl

Jan Mielniczuk

Institute of Computer Science, Polish Academy of Sciences
Ordona 21, 01-237 Warsaw, Poland
Polish-Japanese Institute of Computer Technologies
Koszykowa 86, 02-008 Warszaw, Poland


In the report, the performance of several methods of constructing confidence intervals for a mean of stationary sequence is investigated using extensive simulation study. The studied approaches are sample reuse block methods which do not resort to bootstrap. It turns out that the performance of some known methods strongly depends on a model under consideration and on whether a two-sided or one-sided interval is used. Among the methods studied, the block method based on weak convergence result by Wu (2001) seems to perform most stably.

Keywords:  confidence intervals;  short-range dependence;  reuse block methods;  normal approximation;  iterated random function sequence.

2000 Mathematics Subject Classification: Primary 62G15; Secondary 62G07.


[1] J. Beran, Statistics for Long-Memory Processes, Chapman and Hall, New York 1994.
[2] G. Box and G. Jenkins, Time Series Analysis, Holden Day 1976.
[3] P. Brockwell and R. Davis, Time Series: Theory and Methods, 6th edition, Springer 1998.
[4] J. Ćwik, J. Koronacki and J. Mielniczuk, Testing for a difference between conditional variance functions of nonlinear time series, Control and Cybernetics 29 (2000), 33-50.
[5] E. Carlstein, The use of subseries methods for estimating the variance of a general statistics from a stationary time series, Ann. Statist 14 (1986), 1171-1179.
[6] S. Csörgo, and J. Mielniczuk, Close short-range dependent sums and regression estimation, Acta. Sci. Math. (Szeged) 60 (1995), 177-196.
[7] A. Davison and P. Hall, On studentizing and blocking methods for implementing the bootstrap with dependent data, Austr. J. Statist. 35 (1992), 215-224.
[8] P. Diaconis and D. Freedman, Iterated random functions, SIAM Review 41 (1999), 41-76.
[9] P. Hall and B. Jing, On sample reuse methods for dependent data, J. R. Statist. Soc. B (1996), 727-737.
[10] H.C. Ho and T. Hsing, Limit theorems for functionals of moving averages, Ann. Probab. 25 (1997), 1636-1669.
[11] H. Künsch, The jacknife and the bootstrap for general stationary observations, Ann. Statist. 17 (1989), 1217-1241.
[12] Politis and Romano, Large sample confidence regions based on subsamples under minimal asuumptions, Ann. Statist. 22 (1994), 2031-2050.
[13] M. Rosenblatt, Stationary Sequences and Random Fields, Birkhäuser, Boston 1985.
[14] J. Shao and D. Tu, The Jacknife and Bootstrap, Springer 1995.
[15] K. Singh, On the asymptotic accuracy of Efron's bootstrap, Ann. Statist. 9 (1981), 1187-1195.
[16] W.B. Wu, Studies in time series and random dynamics, Ph. D. thesis, University of Michigan 2001.

Received 9 January 2001