The Annals of Probability

Strong approximation of quantile processes by iterated Kiefer processes

Paul Deheuvels

Full-text: Open access

Abstract

The notion of a $k$th iterated Kiefer process $\mathscr{K}(v,t;k)$ for $k \in \mathbb{N}$ and $v, t \in \mathbb{R}$ is introduced.We show that the uniform quantile process $\beta_n(t)$ may be approximated on [0,1] by $n^{-1/2} \mathscr{K}(n,t;k)$, at an optimal uniform almost sure rate of $O(n^{-1/2 + 1/2^{k+1}+o(1)})$ for each $k \in \mathbb{N}$. Our arguments are based in part on a new functional limit law, of independent interest, for the increments of the empirical process. Applications include an extended version of the uniform Bahadur–Kiefer representation, together with strong limit theorems for nonparametric functional estimators.

Article information

Source
Ann. Probab., Volume 28, Number 2 (2000), 909-945.

Dates
First available in Project Euclid: 18 April 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1019160265

Digital Object Identifier
doi:10.1214/aop/1019160265

Mathematical Reviews number (MathSciNet)
MR1782278

Zentralblatt MATH identifier
1044.60011

Subjects
Primary: 60F05: Central limit and other weak theorems 60F15: Strong theorems 60G15: Gaussian processes 62G30: Order statistics; empirical distribution functions

Keywords
Empirical processes quantile processes order statistics law of the iterated logarithm almost sure convergence strong laws strong invariance principles strong approximation Kiefer processes Wiener process iterated Wiener process iterated Gaussian processes Bahadur–Kiefer-type theorems

Citation

Deheuvels, Paul. Strong approximation of quantile processes by iterated Kiefer processes. Ann. Probab. 28 (2000), no. 2, 909--945. doi:10.1214/aop/1019160265. https://projecteuclid.org/euclid.aop/1019160265


Export citation

References

  • Bahadur, R. (1996). A note on quantiles in large samples. Ann. Math. Statist. 37 577-580.
  • Berkes, I. and Philipp, W. (1979). Approximation theorems for independent and weakly dependent random vectors. Ann. Probab. 7 29-54.
  • Berthet, P. (1997). On the rate of clustering to the Strassen set for increments of the uniform empirical process. J. Theoret. Probab. 10 557-579.
  • Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • Bosq, D. and Lecoutre, J. P. (1987). Th´eorie de l'Estimation Fonctionnelle. Economica, Paris.
  • Burdzy, K. (1993). Some path properties of iterated Brownian motion. In Seminar on Stochastic Processes (E. Çinlar, K. L. Chung and M. Sharpe, eds.) 67-87. Birkh¨auser, Boston.
  • Castelle, N. and Laurent-Bonvalot, F. (1998). Strong approximation of bivariate uniform empirical processes. Ann. Inst. H. Poincar´e 34 425-480.
  • Chung, K. L. (1949). An estimate concerning the Kolmogorov limit distribution. Trans. Amer. Math. Soc. 64 205-233.
  • Cs´aki, E., Cs ¨org o, M., F ¨oldes, A. and R´ev´esz, P. (1989). Brownian local time approximated by a Wiener sheet. Ann. Probab. 17 516-537.
  • Cs´aki, E., Cs ¨org o, M., F ¨oldes, A. and R´ev´esz, P. (1995). Global Strassen-type theorems for iterated Brownian motions. Stochastic Process. Appl. 59 321-341.
  • Cs´aki, E., F ¨oldes, A. and R´ev´esz, P. (1997). Strassen theorems for a class of iterated processes. Trans. Amer. Math. Soc. 349 1153-1167.
  • Cs ¨org o, M. (1983). Quantile Processes with Statistical Applications. SIAM Philadelphia.
  • Cs ¨org o, M., Cs ¨org o, S., Horv´ath, L. and Mason, D. M. (1986). Weighted empirical and quantile processes. Ann. Probab. 14 86-118.
  • Cs ¨org o, M., Cs ¨org o, S., Horv´ath, L. and R´ev´esz, P. (1984). On weak and strong approximations of the quantile processes. In Proceedings of the Seventh Conference on Probability Theory (M. Iosifescu, ed.) 81-95, Editura Academiei, Bucarest.
  • Cs ¨org o, M. and Horv´ath, L. (1986). Approximations of weighted empirical and quantile processes. Statist. Probab. Lett. 4 275-280.
  • Cs ¨org o, M. and Horv´ath, L. (1993). Weighted Approximations in Probability and Statistics. Wiley, New York.
  • Cs ¨org o, M. and R´ev´esz, P. (1975). Some notes on the empirical distribution function and the quantile process. In Limit Theorems of Probability Theory. (P. R´ev´esz, ed.) 11 59-71. North-Holland, Amsterdam.
  • Cs ¨org o, M. and R´ev´esz, P. (1978). Strong approximations of the quantile process. Ann. Statist. 6 882-894.
  • Cs ¨org o, M. and R´ev´esz, P. (1981). Strong Approximations in Probability and Statistics. Academic Press, New York.
  • Cs ¨org o, M. and R´ev´esz, P. (1984). Two approaches to constructing simultaneous confidence bounds for quantiles. Probab. Math. Statist. 4 221-236.
  • Deheuvels, P. (1997). Strong laws for local quantile processes. Ann. Probab. 25 2007-2054.
  • Deheuvels, P. (1998). On the approximation of quantile processes by Kiefer processes. J. Theoret. Probab. 11 997-1018.
  • Deheuvels, P. and Mason, D. M. (1990). Bahadur-Kiefer-type processes. Ann. Probab. 18 669-697. Deheuvels, P. and Mason, D. M. (1992a). A functional LIL approach to pointwise Bahadur- Kiefer theorems. In Probability in Banach Spaces 8 (R. M. Dudley, M. G. Hahn and J. Kuelbs, eds.) 255-266. Birkh¨auser, Boston. Deheuvels, P. and Mason, D. M. (1992b). Functional laws of the iterated logarithm for the increments of empirical and quantile processes. Ann. Probab. 20 1248-1287.
  • Deuschel, J. D. and Stroock, D. W. (1989). Large Deviations. Academic Press, New York.
  • Devroye, L. and Gy ¨orfi, L. (1985). Nonparametric Density Estimation: The L1 View. Wiley, New York.
  • Donsker, M. (1952). Justification and extension of Doob's heuristic approach to the Kolmogorov- Smirnov theorems. Ann. Math. Statist. 23 277-281.
  • Doob, J. L. (1949). Heuristic approach to the Kolmogorov-Smirnov theorems. Ann. Math. Statist. 20 393-403.
  • Finkelstein, H. (1971). The law of the iterated logarithm for empirical distributions. Ann. Math. Statist. 42 607-615.
  • Hu, Y., Pierre-Loti-Viaud, D. and Shi,(1995). Laws of the iterated logarithm for iterated Wiener processes. J. Theoret. Probab. 9 303-319.
  • Khoshnevisan, D. and Lewis, T. M. (1996). Chung's law of the iterated logarithm for iterated Brownian motion. Ann. Inst. H. Poincar´e. 32 349-359.
  • Kiefer, J. (1970). Deviations between the sample quantile process and the sample d.f. In Nonparametric Techniques in Statistical Inference (M. Puri, ed.) 299-319. Cambridge Univ. Press.
  • Kiefer, J. (1972). Skorohod embedding of multivariate rv's and the sample df.Wahrsch. Verw. Gebiete 24 1-35.
  • Koml ´os, J., Major, P. and Tusn´ady, G. (1975). An approximation of partial sums of independent r.v.'s and the sample df. I.Wahrsch. Verw. Gebiete 32 111-131.
  • Koml ´os, J., Major, P. and Tusn´ady, G. (1976). An approximation of partial sums of independent r.v.'s and the sample df. II.Wahrsch. Verw. Gebiete 34 33-58.
  • Schilder, M. (1966). Asymptotic formulas for Wiener integrals. Trans. Amer. Math. Soc. 125 63-85.
  • Scott, D. W. (1992). Multivariate Density Estimation. Wiley, New York.
  • Shorack, G. R. (1982). Kiefer's theorem via the Hungarian construction.Wahrsch. Verw. Gebiete 61 369-373.
  • Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York.
  • Strassen, V. (1964). An invariance principle for the law of the iterated logarithm.Wahrsch. Verw. Gebiete 3 211-226.
  • Stute, W. (1982). The oscillation behaviour of empirical processes. Ann. Probab. 10 86-107.