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On the Bahadur representation of sample quantiles for dependent sequences. (English) Zbl 1080.62024
From the paper: Let \((\varepsilon_k)_{k\in\mathbb{Z}}\) be independent and identically distributed (i.i.d.) random variables and let \(G\) be a measurable function such that \[ X_n=G(\dots,\varepsilon_{n-1}, \varepsilon_n) \] is a well-defined random variable. Clearly \(X_n\) represents a huge class of stationary processes. Let \(F(x)=\mathbb{P}(X_n\leq x)\) be the marginal distribution function of \(X_n\) and let \(f\) be its density. For \(0<p<1\), denote by \(\xi_p=\inf\{x:F(x)\geq p\}\) the \(p\)-th quantile of \(F\). Given a sample \(X_1,\dots,X_n\), let \(\xi_{n,p}\) be the \(p\) th \((0<p<1)\) sample quantile and define the empirical distribution function \[ F_n(x)=n^{-1}\sum^n_{i=1}{\mathbf 1}_{X_i\leq x}. \] For simplicity we also refer to \(\xi_{n,p}\) as the \(p\) th quantile of \(F_n\). We are interested in finding asymptotic representations of \(\xi_{n,p}\).
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.

MSC:
62G20 Asymptotic properties of nonparametric inference
62G30 Order statistics; empirical distribution functions
60F05 Central limit and other weak theorems
60F17 Functional limit theorems; invariance principles
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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References:
[1] Bahadur, R. R. (1966). A note on quantiles in large samples. Ann. Math. Statist. 37 577–580. · Zbl 0147.18805
[2] Billingsley, P. (1968). Convergence of Probability Measures . Wiley, New York. · Zbl 0172.21201
[3] Chow, Y. S. and Teicher, H. (1988). Probability Theory . Independence , Interchangeability , Martingales , 2nd ed. Springer, New York. · Zbl 0652.60001
[4] Deheuvels, P. (1997). Strong laws for local quantile processes. Ann. Probab. 25 2007–2054. · Zbl 0902.60027
[5] 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. JSTOR: · Zbl 0760.60028
[6] Dehling, H., Mikosch, T. and Sørensen, M., eds. (2002). Empirical Process Techniques for Dependent Data . Birkhäuser, Boston. · Zbl 1005.00016
[7] 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. · Zbl 0696.60032
[8] Diaconis, P. and Freedman, D. (1999). Iterated random functions. SIAM Rev. 41 45–76. · Zbl 0926.60056
[9] 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. · Zbl 0948.60012
[10] Einmahl, J. H. J. (1996). A short and elementary proof of the main Bahadur–Kiefer theorem. Ann. Probab. 24 526–531. · Zbl 0897.62046
[11] Elton, J. H. (1990). A multiplicative ergodic theorem for Lipschitz maps. Stochastic Process. Appl. 34 39–47. · Zbl 0686.60028
[12] Freedman, D. A. (1975). On tail probabilities for martingales. Ann. Probab. 3 100–118. · Zbl 0313.60037
[13] Gordin, M. I. (1969). The central limit theorem for stationary processes. Dokl. Akad. Nauk SSSR 188 739–741. · Zbl 0212.50005
[14] Gordin, M. I. and Lifsic, B. (1978). The central limit theorem for stationary Markov processes. Soviet Math. Dokl. 19 392–394. · Zbl 0395.60057
[15] Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application . Academic Press, New York. · Zbl 0462.60045
[16] 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. JSTOR: · Zbl 0712.62042
[17] 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. · Zbl 0862.60026
[18] Hsing, T. and Wu, W. B. (2004). On weighted \(U\)-statistics for stationary processes. Ann. Probab. 32 1600–1631. · Zbl 1049.62099
[19] Kiefer, J. (1967). On Bahadur’s representation of sample quantiles. Ann. Math. Statist. 38 1323–1342. · Zbl 0158.37005
[20] 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.
[21] 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.
[22] Lai, T. L. and Stout, W. (1980). Limit theorems for sums of dependent random variables. Z. Wahrsch. Verw. Gebiete 51 1–14. · Zbl 0419.60026
[23] Major, P. (1981). Multiple Wiener–Itô Integrals : With Applications to Limit Theorems . Springer, Berlin. · Zbl 0451.60002
[24] Móricz, F. (1976). Moment inequalities and the strong laws of large numbers. Z. Wahrsch. Verw. Gebiete 35 299–314. · Zbl 0314.60023
[25] Resnick, S. I. (1987). Extreme Values , Regular Variation and Point Processes . Springer, New York. · Zbl 0633.60001
[26] Sen, P. K. (1968). Asymptotic normality of sample quantiles for \(m\)-dependent processes. Ann. Math. Statist. 39 1724–1730. · Zbl 0197.16102
[27] Sen, P. K. (1972). On the Bahadur representation of sample quantiles for sequences of \(\phi\)-mixing random variables. J. Multivariate Anal. 2 77–95. · Zbl 0226.60050
[28] Serfling, R. J. (1970). Moment inequalities for the maximum cumulative sum. Ann. Math. Statist. 41 1227–1234. · Zbl 0272.60013
[29] Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics . Wiley, New York. · Zbl 1170.62365
[30] Stigler, S. (1973). The asymptotic distribution of the trimmed mean. Ann. Statist. 1 472–477. · Zbl 0261.62016
[31] Wu, W. B. (2003). Empirical processes of long-memory sequences. Bernoulli 9 809–831. · Zbl 1188.62288
[32] Wu, W. B. (2003). Additive functionals of infinite-variance moving averages. Statist. Sinica 13 1259–1267. · Zbl 1045.62092
[33] Wu, W. B. (2004). On strong convergence for sums of stationary processes.
[34] Wu, W. B. and Mielniczuk, J. (2002). Kernel density estimation for linear processes. Ann. Statist. 30 1441–1459. · Zbl 1015.62034
[35] Wu, W. B. and Shao, X. (2004). Limit theorems for iterated random functions. J. Appl. Probab. 41 425–436. · Zbl 1046.60024
[36] Wu, W. B. and Woodroofe, M. (2000). A central limit theorem for iterated random functions. J. Appl. Probab. 37 748–755. · Zbl 0969.60032
[37] Wu, W. B. and Woodroofe, M. (2004). Martingale approximations for sums of stationary processes. Ann. Probab. 32 1674–1690. · Zbl 1057.60022
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