Bernoulli

  • Bernoulli
  • Volume 17, Number 1 (2011), 466-483.

The distribution of the maximal difference between a Brownian bridge and its concave majorant

Fadoua Balabdaoui and Jim Pitman

Full-text: Open access

Abstract

We provide a representation of the maximal difference between a standard Brownian bridge and its concave majorant on the unit interval, from which we deduce expressions for the distribution and density functions and moments of this difference. This maximal difference has an application in nonparametric statistics, where it arises in testing monotonicity of a density or regression curve.

Article information

Source
Bernoulli, Volume 17, Number 1 (2011), 466-483.

Dates
First available in Project Euclid: 8 February 2011

Permanent link to this document
https://projecteuclid.org/euclid.bj/1297173851

Digital Object Identifier
doi:10.3150/10-BEJ280

Mathematical Reviews number (MathSciNet)
MR2798000

Zentralblatt MATH identifier
1284.60151

Keywords
Brownian bridge Brownian excursion concave majorant Doob’s transformation inverse of Laplace transform stick-breaking process

Citation

Balabdaoui, Fadoua; Pitman, Jim. The distribution of the maximal difference between a Brownian bridge and its concave majorant. Bernoulli 17 (2011), no. 1, 466--483. doi:10.3150/10-BEJ280. https://projecteuclid.org/euclid.bj/1297173851


Export citation

References

  • [1] Abate, J. and Whitt, W. (2006). A unified framework for numerically inverting Laplace transforms. INFORMS J. Comput. 18 408–421.
  • [2] Aldous, D. and Pitman, J. (2002). The asymptotic distribution of the diameter of a random mapping. C. R. Acad. Sci. Paris Ser. I 334 1021–1024.
  • [3] Aldous, D. and Pitman, J. (2006). Two recursive decompositions of Brownian bridge related to the asymptotics of random mappings. In In Memoriam Paul-André Meyer: Séminaire de Probabilités XXXIX (M. Émery and M. Yor, eds.). Lecture Notes in Math. 1874 269–303. Berlin: Springer.
  • [4] Bass, R.F. (1984). Markov processes and convex minorants. In Seminar on Probability, XVIII. Lecture Notes in Math. 1059 29–41. Berlin: Springer.
  • [5] Biane, P., Pitman, J. and Yor, M. (2001). Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions. Bull. Amer. Math. Soc. 38 435–465.
  • [6] Blumenthal, R.M. (1983). Weak convergence to Brownian excursion. Ann. Probab. 11 798–800.
  • [7] Çinlar, E. (1992). Sunset over Brownistan. Stochastic Process. Appl. 40 45–53.
  • [8] Durot, C. (2003). A Kolmogorov-type test for monotonicity of regression. Statist. Probab. Lett. 63 425–433.
  • [9] Goldie, C.M. (1989). Records, permutations and greatest convex minorants. Math. Proc. Cambridge Philos. Soc. 106 169–177.
  • [10] Groeneboom, P. (1983). The concave majorant of Brownian motion. Ann. Probab. 11 1016–1027.
  • [11] Hoppe, F.M. (1986). Size-biased filtering of Poisson–Dirichlet samples with an application to partition structures in genetics. J. Appl. Probab. 23 1008–1012.
  • [12] Hunter, D.B. (1964). The calculation of certain Bessel functions. Math. Comp. 18 123–128.
  • [13] Kennedy, D.P. (1976). The distribution of the maximum Brownian excursion. J. Appl. Probab. 13 371–376.
  • [14] Kulikov, V.N. and Lopuhaä, H.P. (2005). Asymptotic normality of the Lk-error of the Grenander estimator. Ann. Statist. 33 2228–2255.
  • [15] Kulikov, V.N. and Lopuhaä, H.P. (2008). Distribution of global measures of deviation between the empirical distribution function and its concave majorant. J. Theoret. Probab. 21 356–377.
  • [16] Mechel, F. (1966). Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 407–412.
  • [17] Pitman, J.W. (1983). Remarks on the convex minorant of Brownian motion. In Seminar on Stochastic Processes, 1982 (Evanston, Ill., 1982). Progr. Probab. Statist. 5 219–227. Boston, MA: Birkhäuser.
  • [18] Pitman, J. (1996). Random discrete distributions invariant under size-biased permutation. Adv. in Appl. Probab. 28 525–539.
  • [19] Pitman, J. and Yor, M. (1997). The two-parameter Poisson–Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25 855–900.
  • [20] Pitman, J. and Yor, M. (2001). On the distribution of ranked heights of excursions of a Brownian bridge. Ann. Probab. 29 361–384.
  • [21] Rosenbaum, S. (1961). Moments of a truncated bivariate normal distribution. J. Roy. Statist. Soc. Ser. B 23 405–408.
  • [22] Steele, J.M. (2002). The Bohnenblust–Spitzer algorithm and its applications. J. Comput. Appl. Math. 142 235–249.
  • [23] Stuart, A. and Ord, J.K. (1998). Kendall’s Advanced Theory of Statistics, Vol. 1: Distribution Theory. New York: Oxford Univ. Press.
  • [24] Suidan, T.M. (2001). Convex minorants of random walks and Brownian motion. Teor. Veroyatn. Primen. 46 498–512.
  • [25] Vershik, A.M. and Shmidt, A.A. (1977). Limit measures arising in the asymptotic theory of symmetric groups. I. Theory Probab. Appl. 22 70–85.