The Annals of Statistics

On the Bahadur representation of sample quantiles for dependent sequences

Wei Biao Wu

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Abstract

We establish the Bahadur representation of sample quantiles for linear and some widely used nonlinear processes. Local fluctuations of empirical processes are discussed. Applications to the trimmed and Winsorized means are given. Our results extend previous ones by establishing sharper bounds under milder conditions and thus provide new insight into the theory of empirical processes for dependent random variables.

Article information

Source
Ann. Statist., Volume 33, Number 4 (2005), 1934-1963.

Dates
First available in Project Euclid: 5 August 2005

Permanent link to this document
https://projecteuclid.org/euclid.aos/1123250233

Digital Object Identifier
doi:10.1214/009053605000000291

Mathematical Reviews number (MathSciNet)
MR2166566

Zentralblatt MATH identifier
1080.62024

Subjects
Primary: 62G30: Order statistics; empirical distribution functions 60F05: Central limit and other weak theorems
Secondary: 60F17: Functional limit theorems; invariance principles

Keywords
Long- and short-range dependence Bahadur representation nonlinear time series almost sure convergence linear process martingale inequalities empirical processes law of the iterated logarithm

Citation

Wu, Wei Biao. On the Bahadur representation of sample quantiles for dependent sequences. Ann. Statist. 33 (2005), no. 4, 1934--1963. doi:10.1214/009053605000000291. https://projecteuclid.org/euclid.aos/1123250233


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References

  • Bahadur, R. R. (1966). A note on quantiles in large samples. Ann. Math. Statist. 37 577--580.
  • Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • Chow, Y. S. and Teicher, H. (1988). Probability Theory. Independence, Interchangeability, Martingales, 2nd ed. Springer, New York.
  • Deheuvels, P. (1997). Strong laws for local quantile processes. Ann. Probab. 25 2007--2054.
  • Deheuvels, P. and Mason, D. M. (1992). Functional laws of the iterated logarithm for the increments of empirical and quantile processes. Ann. Probab. 20 1248--1287.
  • Dehling, H., Mikosch, T. and Sørensen, M., eds. (2002). Empirical Process Techniques for Dependent Data. Birkhäuser, Boston.
  • Dehling, H. and Taqqu, M. (1989). The empirical process of some long-range dependent sequences with an application to $U$-statistics. Ann. Statist. 17 1767--1783.
  • Diaconis, P. and Freedman, D. (1999). Iterated random functions. SIAM Rev. 41 45--76.
  • Doukhan, P. and Surgailis, D. (1998). Functional central limit theorem for the empirical process of short memory linear processes. C. R. Acad. Sci. Paris Sér. I Math. 326 87--92.
  • Einmahl, J. H. J. (1996). A short and elementary proof of the main Bahadur--Kiefer theorem. Ann. Probab. 24 526--531.
  • Elton, J. H. (1990). A multiplicative ergodic theorem for Lipschitz maps. Stochastic Process. Appl. 34 39--47.
  • Freedman, D. A. (1975). On tail probabilities for martingales. Ann. Probab. 3 100--118.
  • Gordin, M. I. (1969). The central limit theorem for stationary processes. Dokl. Akad. Nauk SSSR 188 739--741.
  • Gordin, M. I. and Lifsic, B. (1978). The central limit theorem for stationary Markov processes. Soviet Math. Dokl. 19 392--394.
  • Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York.
  • Hesse, C. H. (1990). A Bahadur-type representation for empirical quantiles of a large class of stationary, possibly infinite-variance, linear processes. Ann. Statist. 18 1188--1202.
  • Ho, H.-C. and Hsing, T. (1996). On the asymptotic expansion of the empirical process of long-memory moving averages. Ann. Statist. 24 992--1024.
  • Hsing, T. and Wu, W. B. (2004). On weighted $U$-statistics for stationary processes. Ann. Probab. 32 1600--1631.
  • Kiefer, J. (1967). On Bahadur's representation of sample quantiles. Ann. Math. Statist. 38 1323--1342.
  • Kiefer, J. (1970). Deviations between the sample quantile process and the sample df. In Nonparametric Techniques in Statistical Inference (M. L. Puri, ed.) 299--319. Cambridge Univ. Press, London.
  • Kiefer, J. (1970). Old and new methods for studying order statistics and sample quantiles. In Nonparametric Techniques in Statistical Inference (M. L. Puri, ed.) 349--357. Cambridge Univ. Press, London.
  • Lai, T. L. and Stout, W. (1980). Limit theorems for sums of dependent random variables. Z. Wahrsch. Verw. Gebiete 51 1--14.
  • Major, P. (1981). Multiple Wiener--Itô Integrals: With Applications to Limit Theorems. Springer, Berlin.
  • Móricz, F. (1976). Moment inequalities and the strong laws of large numbers. Z. Wahrsch. Verw. Gebiete 35 299--314.
  • Resnick, S. I. (1987). Extreme Values, Regular Variation and Point Processes. Springer, New York.
  • Sen, P. K. (1968). Asymptotic normality of sample quantiles for $m$-dependent processes. Ann. Math. Statist. 39 1724--1730.
  • Sen, P. K. (1972). On the Bahadur representation of sample quantiles for sequences of $\phi$-mixing random variables. J. Multivariate Anal. 2 77--95.
  • Serfling, R. J. (1970). Moment inequalities for the maximum cumulative sum. Ann. Math. Statist. 41 1227--1234.
  • Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York.
  • Stigler, S. (1973). The asymptotic distribution of the trimmed mean. Ann. Statist. 1 472--477.
  • Wu, W. B. (2003). Empirical processes of long-memory sequences. Bernoulli 9 809--831.
  • Wu, W. B. (2003). Additive functionals of infinite-variance moving averages. Statist. Sinica 13 1259--1267.
  • Wu, W. B. (2004). On strong convergence for sums of stationary processes. Preprint.
  • Wu, W. B. and Mielniczuk, J. (2002). Kernel density estimation for linear processes. Ann. Statist. 30 1441--1459.
  • Wu, W. B. and Shao, X. (2004). Limit theorems for iterated random functions. J. Appl. Probab. 41 425--436.
  • Wu, W. B. and Woodroofe, M. (2000). A central limit theorem for iterated random functions. J. Appl. Probab. 37 748--755.
  • Wu, W. B. and Woodroofe, M. (2004). Martingale approximations for sums of stationary processes. Ann. Probab. 32 1674--1690.