Summary
Let {W(t): t ∈ ∝} be two-sided Brownian motion, originating from zero, and let V(a) be defined by V(a)=sup}t ∈ ∝: W(t)−(t−a)2 is maximal}. Then {V(a): a ∈ ℝ} is a Markovian jump process, running through the locations of maxima of two-sided Brownian motion with respect to the parabolas f a(t)=(t−a)2. We give an analytic expression for the infinitesimal generators of the processes a ∈ ℝ, in terms of Airy functions in Theorem 4.1. This makes it possible to develop asymptotics for the global behavior of a large class of isotonic estimators (i.e. estimators derived under order restrictions). An example of this is given in Groeneboom (1985), where the asymptotic distribution of the (standardized) L 1-distance between a decreasing density and the Grenander maximum likelihood estimator of this density is determined. On our way to Theorem 4.1 we derive some other results. For example, we give an analytic expression for the joint density of the maximum and the location of the maximum of the process {W(t)−ct 2: t ∈ ℝ}, where c is an aribrary positive constant. We also determine the Laplace transform of the integral over a Brownian excursion. These last results also have recently been derived by several other authors, using a variety of methods.
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Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions. Nat. Bur. Stand. 55, Washington (1964)
Barbour, A.D.: Brownian motion and a sharply curved boundary. Adv. Appl. Probab. 13, 736–750 (1981)
Blumenthal, R.M.: Weak convergence to Brownian excursion. Ann. Probab. 11, 798–800 (1983)
Chernoff, H.: Estimation of the mode. Ann. Inst. Stat. Math. 16, 31–41 (1964)
Daniels, H.E.: The statistical theory of the strength of bundles of threads, I. Proc. Roy. Soc. Lond. Ser. A 183, 404–435 (1945)
Daniels, H.E.: The maximum size of a closed epidemic. Adv. Appl. Probab. 6, 607–621 (1974)
Daniels, H.E., Skyrme, T.H.R.: The maximum of a random walk whose mean path has a maximum. Adv. Appl. Probab. 17, 85–99 (1985)
Dieudonné, J.: Foundations of modern analysis. New York: Academic Press 1969
Doob, J.L.: Classical potential theory and its probabilistic counterpart. Berlin Heidelberg New York Tokyo: Springer 1984
Durrett, R. T., Iglehart, D. L., Miller, D. R.: Weak convergence to Brownian meander and Brownian excursion. Ann. Probab. 5, 117–129 (1977)
Grenander, U.: On the theory of mortality measurement. Part II. Skand. Akt. 39, 125–153 (1956)
Groeneboom, P.: The concave majorant of Brownian motion. Ann. Probab. 11, 1016–1027 (1983)
Groeneboom, P.: Estimating a monotone density. In: Le Cam, L.E., Olshen, R.A. (eds.) Proceedings of the Berkeley conferende in honor of Jerzy Neyman and Jack Kiefer, vol. II, pp. 539–555. Monterey: Wadsworth 1985
Houwelingen, J.C. van: Monotone empirical Bayes tests for uniform distributions using the maximum likelihood estimator of a decreasing density. Ann. Stat. 15, 875–879 (1987)
Itô, K., McKean, H.P. Jr.: Diffusion processes and their sample paths, 2nd ed. Berlin Heidelberg New York: Springer 1974
Kiefer, J., Wolfowitz, J.: Asymptotically minimax estimation of concave and convex distribution functions. Z. Wahrscheinlichkeitstheor. Verw. Geb. 34, 73–85 (1976)
Louchard, G.: Kac's formula, Lévy's local time and Brownian excursion. J. Appl. Probab. 21, 479–499 (1984)
Meyer, P.A., Smythe, R.T., Walsh, J.B.: Birth and death of Markov processes. Sixth Berkeley Symposium 3, 295–305. Berkeley: University of California Press 1972
Olver, F.W.J.: Asymptotics and special functions. New York: Academic Press 1974
Phoenix, S.L., Taylor, H.M.: The asymptotic strength distribution of a general fiber bundle. Adv. Appl. Probab. 5, 200–216 (1973)
Prakasa Rao, B.L.S.: Estimation of a unimodal density. Sankhya Ser. A 31, 23–36 (1969)
Shepp, L.A.: On the integral of the absolute value of the pinned Wiener process. Ann. Probab. 10, 234–239 (1982)
Smith, R.L.: The asymptotic distribution of aseries-parallel system with equal load sharing. Ann. Probab. 10, 137–171 (1982)
Stroock, D.W., Varadhan, S.R.S.: Multidimensional diffusion processes. Berlin Heidelberg New York: Springer 1979
Temme, N.M.: A convolution integral equation solved by Laplace transformation. J. Comp. Appl. Math. 13, 609–613 (1985)
Venter, J.H.: On estimation of the mode. Ann. Math. Stat. 38, 1446–1456 (1967)
Williams, D.: Path decomposition and continuity of local time for one-dimensional diffusions. Proc. London Math. Soc. 28, 738–768 (1974)
Williams, D.: Diffusions, Markov processes and martingales. vol. 1: foundations. Chichester New York Brisbane Toronto: Wiley 1979
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This paper was awarded the Rollo Davidson prize 1985 (Cambridge, UK)
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Groeneboom, P. Brownian motion with a parabolic drift and airy functions. Probab. Th. Rel. Fields 81, 79–109 (1989). https://doi.org/10.1007/BF00343738
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DOI: https://doi.org/10.1007/BF00343738