## Chernoff’s density is log-concave.(English)Zbl 1294.60100

Chernoff density, i.e., the density of argmax$$(W(t) - t^2)$$ for a two-sided Brownian motion $$W$$, is shown to be log-concave. A stronger form of log-concavity is conjectured and a partial proof thereof is provided. The proof uses a characterization of Pólya frequency functions due to Schoenberg and a representation for Airy functions due to Merkes and Salmassi.

### MSC:

 60J65 Brownian motion 60E99 Distribution theory

DLMF
Full Text:

### References:

 [1] Ayer, M., Brunk, H.D., Ewing, G.M., Reid, W.T. and Silverman, E. (1955). An empirical distribution function for sampling with incomplete information. Ann. Math. Statist. 26 641-647. · Zbl 0066.38502 [2] Balabdaoui, F., Rufibach, K. and Wellner, J.A. (2009). Limit distribution theory for maximum likelihood estimation of a log-concave density. Ann. Statist. 37 1299-1331. · Zbl 1160.62008 [3] Balabdaoui, F. and Wellner, J.A. (2012). Chernoff’s distribution is log-concave. Technical Report No. tr595.pdf. Dept. Statistics, Univ. Washington. Available at . · Zbl 1294.60100 [4] Barthe, F. (2006). The Brunn-Minkowski theorem and related geometric and functional inequalities. In International Congress of Mathematicians. Vol. II 1529-1546. Zürich: Eur. Math. Soc. · Zbl 1099.39017 [5] Bondesson, L. (1992). Generalized Gamma Convolutions and Related Classes of Distributions and Densities. Lecture Notes in Statistics 76 . New York: Springer. · Zbl 0756.60015 [6] Bondesson, L. (1997). On hyperbolically monotone densities. In Advances in the Theory and Practice of Statistics. Wiley Ser. Probab. Statist. Appl. Probab. Statist. 299-313. New York: Wiley. · Zbl 0887.62014 [7] Brunk, H.D. (1970). Estimation of isotonic regression. In Nonparametric Techniques in Statistical Inference ( Proc. Sympos. , Indiana Univ. , Bloomington , Ind. , 1969) 177-197. London: Cambridge Univ. Press. [8] Caffarelli, L.A. (2000). Monotonicity properties of optimal transportation and the FKG and related inequalities. Comm. Math. Phys. 214 547-563. · Zbl 0978.60107 [9] Chernoff, H. (1964). Estimation of the mode. Ann. Inst. Statist. Math. 16 31-41. · Zbl 0212.21802 [10] Daniels, H.E. and Skyrme, T.H.R. (1985). The maximum of a random walk whose mean path has a maximum. Adv. in Appl. Probab. 17 85-99. · Zbl 0552.60067 [11] Dehling, H. and Philipp, W. (2002). Empirical Process Techniques for Dependent Data . Boston, MA: Birkhäuser. · Zbl 1021.62036 [12] Grenander, U. (1956). On the theory of mortality measurement. I. Skand. Aktuarietidskr. 39 70-96. · Zbl 0073.15404 [13] Grenander, U. (1956). On the theory of mortality measurement. II. Skand. Aktuarietidskr. 39 125-153. · Zbl 0077.33715 [14] Groeneboom, P. (1985). Estimating a monotone density. In Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer , Vol. II ( Berkeley , CA , 1983). Wadsworth Statist./Probab. Ser. 539-555. Belmont, CA: Wadsworth. · Zbl 1373.62144 [15] Groeneboom, P. (1989). Brownian motion with a parabolic drift and Airy functions. Probab. Theory Related Fields 81 79-109. [16] Groeneboom, P. (1996). Lectures on inverse problems. In Lectures on Probability Theory and Statistics ( Saint-Flour , 1994). Lecture Notes in Math. 1648 67-164. Berlin: Springer. · Zbl 0907.62042 [17] Groeneboom, P. (2010). The maximum of Brownian motion minus a parabola. Electron. J. Probab. 15 1930-1937. · Zbl 1226.60110 [18] Groeneboom, P. (2011). Vertices of the least concave majorant of Brownian motion with parabolic drift. Electron. J. Probab. 16 2234-2258. · Zbl 1246.60106 [19] Groeneboom, P., Jongbloed, G. and Wellner, J.A. (2001). A canonical process for estimation of convex functions: The “invelope” of integrated Brownian motion $$+t_{4}$$. Ann. Statist. 29 1620-1652. · Zbl 1043.62026 [20] Groeneboom, P., Jongbloed, G. and Wellner, J.A. (2001). Estimation of a convex function: Characterizations and asymptotic theory. Ann. Statist. 29 1653-1698. · Zbl 1043.62027 [21] Groeneboom, P. and Wellner, J.A. (1992). Information Bounds and Nonparametric Maximum Likelihood Estimation. DMV Seminar 19 . Basel: Birkhäuser. · Zbl 0757.62017 [22] Groeneboom, P. and Wellner, J.A. (2001). Computing Chernoff’s distribution. J. Comput. Graph. Statist. 10 388-400. [23] Hargé, G. (2004). A convex/log-concave correlation inequality for Gaussian measure and an application to abstract Wiener spaces. Probab. Theory Related Fields 130 415-440. · Zbl 1059.60022 [24] Harrison, P.G. (1990). Laplace transform inversion and passage-time distributions in Markov processes. J. Appl. Probab. 27 74-87. · Zbl 0704.60094 [25] Huang, J. and Wellner, J.A. (1995). Estimation of a monotone density or monotone hazard under random censoring. Scand. J. Stat. 22 3-33. · Zbl 0827.62032 [26] Huang, Y. and Zhang, C.H. (1994). Estimating a monotone density from censored observations. Ann. Statist. 22 1256-1274. · Zbl 0821.62016 [27] Janson, S., Louchard, G. and Martin-Löf, A. (2010). The maximum of Brownian motion with parabolic drift. Electron. J. Probab. 15 1893-1929. · Zbl 1226.60111 [28] Karlin, S. (1968). Total Positivity. Vol. I . Stanford, CA: Stanford Univ. Press. · Zbl 0219.47030 [29] Kim, J. and Pollard, D. (1990). Cube root asymptotics. Ann. Statist. 18 191-219. · Zbl 0703.62063 [30] Le Cam, L. (1986). The central limit theorem around 1935. Statist. Sci. 1 78-96. · Zbl 0603.60001 [31] Leurgans, S. (1982). Asymptotic distributions of slope-of-greatest-convex-minorant estimators. Ann. Statist. 10 287-296. · Zbl 0484.62033 [32] Marshall, A.W. and Olkin, I. (1979). Inequalities : Theory of Majorization and Its Applications. Mathematics in Science and Engineering 143 . New York: Academic Press [Harcourt Brace Jovanovich Publishers]. · Zbl 0437.26007 [33] Marshall, A.W., Olkin, I. and Arnold, B.C. (2011). Inequalities : Theory of Majorization and Its Applications , 2nd ed. Springer Series in Statistics . New York: Springer. · Zbl 1219.26003 [34] Merkes, E.P. and Salmassi, M. (1997). On univalence of certain infinite products. Complex Variables Theory Appl. 33 207-215. · Zbl 0907.30020 [35] Olkin, I. and Tong, Y.L. (1988). Peakedness in multivariate distributions. In Statistical Decision Theory and Related Topics , IV , Vol. 2 ( West Lafayette , IN , 1986) 373-383. New York: Springer. · Zbl 0678.62057 [36] Olver, F.W.J., Lozier, D.W., Boisvert, R.F. and Clark, C.W. (2010). NIST Handbook of Mathematical Functions . Washington, DC: U.S. Department of Commerce National Institute of Standards and Technology. · Zbl 1198.00002 [37] Prakasa Rao, B.L.S. (1969). Estkmation of a unimodal density. Sankhyā Ser. A 31 23-36. · Zbl 0181.45901 [38] Prakasa Rao, B.L.S. (1970). Estimation for distributions with monotone failure rate. Ann. Math. Statist. 41 507-519. · Zbl 0214.45903 [39] Proschan, F. (1965). Peakedness of distributions of convex combinations. Ann. Math. Statist. 36 1703-1706. · Zbl 0138.41104 [40] Rockafellar, R.T. and Wets, R.J.B. (1998). Variational Analysis. Grundlehren der Mathematischen Wissenschaften [ Fundamental Principles of Mathematical Sciences ] 317 . Berlin: Springer. [41] Salmassi, M. (1999). Inequalities satisfied by the Airy functions. J. Math. Anal. Appl. 240 574-582. · Zbl 0946.33009 [42] Schoenberg, I.J. (1951). On Pólya frequency functions. I. The totally positive functions and their Laplace transforms. J. Anal. Math. 1 331-374. · Zbl 0045.37602 [43] Shorack, G.R. (2000). Probability for Statisticians. Springer Texts in Statistics . New York: Springer. · Zbl 0951.62005 [44] van Eeden, C. (1957). Maximum likelihood estimation of partially or completely ordered parameters. I. Nederl. Akad. Wetensch. Proc. Ser. A. 60 = Indag. Math. 19 128-136. · Zbl 0086.12803 [45] van Zwet, W.R. (1964). Convex transformations: A new approach to skewness and kurtosis. Stat. Neerl. 18 433-441. [46] van Zwet, W.R. (1964). Convex Transformations of Random Variables. Mathematical Centre Tracts 7 . Amsterdam: Mathematisch Centrum. · Zbl 0125.37102 [47] Wellner, J.A. (2013). Strong log-concavity is preserved by convolution. In High Dimensional Probability VI : The Banff Volume ( Progress in Probability ) (C. Houdré, D. M. Mason, J. Rosiński and J. A. Wellner, eds.) 95-103. Basel: Birkhauser. · Zbl 1271.60034
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