Marginal densities of the least concave majorant of Brownian motion. (English) Zbl 1044.60072

Summary: A clean, closed form, joint density is derived for Brownian motion, its least concave majorant, and its derivative, all at the same fixed point. Some remarkable conditional and marginal distributions follow from this joint density. For example, it is shown that the height of the least concave majorant of Brownian motion at a fixed time point has the same distribution as the distance from the Brownian motion path to its least concave majorant at the same fixed time point. Also, it is shown that conditional on the height of the least concave majorant of Brownian motion at a fixed time point, the left-hand slope of the least concave majorant of Brownian motion at the same fixed time point is uniformly distributed.


60J65 Brownian motion
62E15 Exact distribution theory in statistics
62H10 Multivariate distribution of statistics
Full Text: DOI


[1] Barlow, R. E., Bartholomew, D. J., Bremner, J. M. and Brunk, H. D. (1972). Statistical Inference under Order Restrictions. Wiley, London. · Zbl 0246.62038
[2] Bass, R. F. (1983). Markov processes and convex minorants. Seminaire de Probabilities XVIII. Lecture Notes in Math. 1059 29-41. Springer, Berlin. · Zbl 0575.60080
[3] Brunk, H. D., Franck, W. E., Hanson, D. L. and Hogg, R. V. (1966). Maximum likelihood estimation of the distributions of two stochastically ordered random variables. J. Amer. Amer. Statist. Assoc. 61 1067-1080. JSTOR: · Zbl 0146.40101
[4] Dykstra, R. (1982). Maximum likelihood estimation of survival functions of stochastically ordered random variables. J. Amer. Amer. Statist. Assoc. 77 621-628. JSTOR: · Zbl 0498.62040
[5] Dykstra, R., Kochar, S. and Robertson, T. (1995). Inference for likelihood ratio ordering in the two-sample problem. J. Amer. Amer. Statist. Assoc. 90 1034-1040. JSTOR: · Zbl 0843.62032
[6] Grenander, U. (1956). On the theory of mortality measurement. Part II. Skand. Akt. 39 125-153. · Zbl 0073.15404
[7] Groeneboom, P. (1983). The concave majorant of Brownian motion. Ann. Probab. 11 1016-1027. · Zbl 0523.60079
[8] Groeneboom, P. (1985). Estimating a monotone density. In Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer (L. M. LeCam and R. A. Olshen, eds.) 2 529-555. Wadsworth, Belmont, CA. · Zbl 1373.62144
[9] It o, K. and McKean, H. P. Jr. (1974). Diffusion prossesses and Their Sample Paths, 2nd ed. Springer, Berlin.
[10] Marshall, A. W. and Proschan, F. (1965). Maximum likelihood estimation for distributions with monotone failure rates. Ann. Math. Statist. 36 69-77. · Zbl 0128.38506
[11] Pitman, J. W. (1983). Remarks on the convex minorant of Brownian motion. In Seminar on Stochastic Processes (E. \?Cinlar, K. L. Chung and R. K. Getoor, eds.) 219-228. Birkhäuser, Boston. · Zbl 0528.60033
[12] Praestgaard, J. T. and Huang, J. (1996). Asymptotic theory for Nnnparametric estimation of survival curves under order restrictions. Ann. Statist. 24 1679-1716. · Zbl 0896.62044
[13] Prakasa Rao, B. L. S. (1969). Estimation of a unimodal density. Sankhya Ser. A 31 23-36. · Zbl 0181.45901
[14] Robertson, T. and Wright, F. T. (1981). Likelihood ratio tests for and against stochastic ordering between multinomial populations. Ann. Statist. 9 1248-57. · Zbl 0474.62038
[15] Robertson, T., Wright, F. T. and Dykstra, R. L. (1988). Order Restricted Inference. Wiley, New York. · Zbl 0645.62028
[16] Williams, D. (1974). Path decomposition and continuity of local time for one-dimensional diffusions, I. Proc. London Math. Soc. Ser. 3 28 738-768. · Zbl 0326.60093
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