A short and elementary proof of the main Bahadur-Kiefer theorem

A short and elementary proof of the main Bahadur-Kiefer theorem

John H. J. Einmahl

Source: Ann. Probab. Volume 24, Number 1 (1996), 526-531.

Abstract

A short proof of the lower bound in the strong version of the famous Theorem 1A in Kiefer (1970) on the Bahadur-Kiefer process is presented. The proof is elementary and, in particular, does not use strong approximations.

First Page: Show

Primary Subjects: 62G30, 60F15

Keywords: Bahadur-Kiefer process; empirical and quantile process; strong law

Full-text: Open access

PDF File (166 KB)

Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aop/1042644730
Mathematical Reviews number (MathSciNet): MR1387649
Digital Object Identifier: doi:10.1214/aop/1042644730
Zentralblatt MATH identifier: 0897.62046

back to Table of Contents

References

BAHADUR, R. R. 1966. A note on quantiles in large samples. Ann. Math. Statist. 37 577 580. Z.

Mathematical Reviews (MathSciNet): MR32:6522

Zentralblatt MATH: 0147.18805

Digital Object Identifier: doi:10.1214/aoms/1177699450

Project Euclid: euclid.aoms/1177699450

DEHEUVELS, P. and MASON, D. M. 1990. Bahadur Kiefer-ty pe processes. Ann. Probab. 18 669 697. Z.

Mathematical Reviews (MathSciNet): MR91j:60055

Zentralblatt MATH: 0712.60028

Digital Object Identifier: doi:10.1214/aop/1176990852

Project Euclid: euclid.aop/1176990852

DVORETZKY, A., KIEFER, J. C. and WOLFOWITZ, J. 1956. Asy mptotic minimax character of the sample distribution function and of the classical multinomial estimator. Ann. Math. Statist. 27 642 669. Z.

Mathematical Reviews (MathSciNet): MR18,772i

Zentralblatt MATH: 0073.14603

Digital Object Identifier: doi:10.1214/aoms/1177728174

Project Euclid: euclid.aoms/1177728174

KIEFER, J. C. 1970. Deviations between the sample quantile process and the sample df. In Z. Non-parametric Techniques in Statistical Inference M. Puri, ed. 299 319. Cambridge Univ. Press. ¨ Z.

Mathematical Reviews (MathSciNet): MR43:2808

KOLMOGOROV, A. N. 1929. Uber das Gesetz des iterierten Logarithmus. Math. Ann. 101 126 135.Z.

Mathematical Reviews (MathSciNet): MR1512520

Digital Object Identifier: doi:10.1007/BF01454828

MALLOWS, C. L. 1968. An inequality involving multinomial probabilities. Biometrika 55 422 424. Z.

Zentralblatt MATH: 0162.50707

MASON, D. M., SHORACK, G. R. and WELLNER, J. A. 1983. Strong limit theorems for oscillation moduli of the uniform empirical process. Z. Wahrsch. Verw. Gebiete 65 83 97. Z.

Zentralblatt MATH: 0506.60025

Mathematical Reviews (MathSciNet): MR717935

MASSART, P. 1990. The tight constant in the Dvoretzky Kiefer Wolfowitz inequality. Ann. Probab. 18 1269 1283. Z. MOGUL'SKII, A. A. 1979. On the law of the iterated logarithm in Chung's form for functional spaces. Theory Probab. Appl. 24 405 413. Z.

Mathematical Reviews (MathSciNet): MR1062069

Zentralblatt MATH: 0713.62021

Digital Object Identifier: doi:10.1214/aop/1176990746

Project Euclid: euclid.aop/1176990746

SHORACK, G. R. 1982. Kiefer's theorem via the Hungarian construction. Z. Wahrsch. Verw. Gebiete 61 369 373. Z.

Mathematical Reviews (MathSciNet): MR84b:60037

Zentralblatt MATH: 0501.60038

Digital Object Identifier: doi:10.1007/BF00539836

SHORACK, G. R. and WELLNER, J. A. 1986. Empirical Processes with Applications to Statistics. Wiley, New York. Z.

Mathematical Reviews (MathSciNet): MR88e:60002

STUTE, W. 1982. The oscillation behavior of empirical processes. Ann. Probab. 10 86 107.

Mathematical Reviews (MathSciNet): MR83e:60028