The Annals of Statistics

Tusnády’s inequality revisited

Andrew Carter and David Pollard

Full-text: Open access

Abstract

Tusnády’s inequality is the key ingredient in the KMT/Hungarian coupling of the empirical distribution function with a Brownian bridge. We present an elementary proof of a result that sharpens the Tusnády inequality, modulo constants. Our method uses the beta integral representation of Binomial tails, simple Taylor expansion and some novel bounds for the ratios of normal tail probabilities.

Article information

Source
Ann. Statist., Volume 32, Number 6 (2004), 2731-2741.

Dates
First available in Project Euclid: 7 February 2005

Permanent link to this document
https://projecteuclid.org/euclid.aos/1107794885

Digital Object Identifier
doi:10.1214/009053604000000733

Mathematical Reviews number (MathSciNet)
MR2154001

Zentralblatt MATH identifier
1076.62012

Subjects
Primary: 62E17: Approximations to distributions (nonasymptotic)
Secondary: 62B15: Theory of statistical experiments

Keywords
Quantile coupling KMT/Hungarian construction Tusnády’s inequality beta integral representation of Binomial tails ratios of normal tails equivalent normal deviate

Citation

Carter, Andrew; Pollard, David. Tusnády’s inequality revisited. Ann. Statist. 32 (2004), no. 6, 2731--2741. doi:10.1214/009053604000000733. https://projecteuclid.org/euclid.aos/1107794885


Export citation

References

  • Bretagnolle, J. and Massart, P. (1989). Hungarian constructions from the nonasymptotic viewpoint. Ann. Probab. 17 239–256.
  • Brown, L. D., Carter, A. V., Low, M. G. and Zhang, C.-H. (2004). Equivalence theory for density estimation, Poisson processes and Gaussian white noise with drift. Ann. Statist. 32 2074–2097.
  • Csörgő, M. and Révész, P. (1981). Strong Approximations in Probability and Statistics. Academic Press, New York.
  • de Bruijn, N. G. (1981). Asymptotic Methods in Analysis, corrected 3rd ed. Dover, New York.
  • Dudley, R. M. (2000). Notes on empirical processes. Lecture notes for a course given at Aarhus Univ., August 1999.
  • Feller, W. (1968). An Introduction to Probability Theory and Its Applications 1, 3rd ed. Wiley, New York.
  • Komlós, J., Major, P. and Tusnády, G. (1975). An approximation of partial sums of independent rv's and the sample df. I. Z. Wahrsch. Verw. Gebiete 32 111–131.
  • Major, P. (2000). The approximation of the normalized empirical ditribution function by a Brownian bridge. Technical report, Mathematical Institute of the Hungarian Academy of Sciences. Notes available from www.renyi.hu/~major/.
  • Mason, D. M. (2001). Notes on the KMT Brownian bridge approximation to the uniform empirical process. In Asymptotic Methods in Probability and Statistics with Applications (N. Balakrishnan, I. A. Ibragimov and V. B. Nevzorov, eds.) 351–369. Birkhäuser, Boston.
  • Massart, P. (2002). Tusnády's lemma, 24 years later. Ann. Inst. H. Poincaré Probab. Statist. 38 991–1007.
  • Molenaar, W. (1970). Approximations to the Poisson, Binomial, and Hypergeometric Distribution Functions. Math. Centrum, Amsterdam.
  • Nussbaum, M. (1996). Asymptotic equivalence of density estimation and Gaussian white noise. Ann. Statist. 24 2399–2430.
  • Peizer, D. B. and Pratt, J. W. (1968). A normal approximation for Binomial, F, beta, and other common, related tail probabilities. I. J. Amer. Statist. Assoc. 63 1416–1456.
  • Petrov, V. V. (1975). Sums of Independent Random Variables. Springer, New York.
  • Pratt, J. W. (1968). A normal approximation for Binomial, F, beta, and other common, related tail probabilities. II. J. Amer. Statist. Assoc. 63 1457–1483.
  • Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York.
  • Tusnády, G. (1977). A study of statistical hypotheses. Ph. D. thesis, Hungarian Academy of Sciences, Budapest.