Jon Wellner, Chernoff's Distribution is log-concave. But why? (And why does it matter?)
Abstract: Chernoff's distribution, which apparently arose first in connection with nonparametric estimation of the mode of a uni-modal density, is the density of the location of the maximum of two-sided Brownian motion minus a parabola. By Groeneboom's switching relation this random variable has the same distribution as the slope at zero of the least concave majorant of two-sided Brownian motion minus a parabola, and it is in this connection that it arises naturally as the limit distribution (up to multiplicative constants) of nonparametric estimators of monotone functions. It was studied further by Daniels and Skryme (1985) and by Groeneboom (1985), (1989), and computed by Groeneboom and Wellner (2001). It appears “Gaussian" in shape; and it is natu-ral to conjecture that Chernoff's distribution is log-concave.
In this talk I will review some results concerning log-concave distributions on one and higher dimensional Euclidean spaces, with emphasis on preservation results and connections with questions arising from statistics. I will indicate why Chernoff's distribution is log-concave, and briefly mention further problems.
Some background material for this talk is given in the book “Unimodality, Convexity and Applications" by S. Dharmadhikari and K. Joag-dev, Academic Press, 1988, and in the papers entioned above.