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

The Concave Majorant of Brownian Motion

A Berry-Esseen Type Bound for a Smoothed Version of Grenander Estimator

In various statistical model, such as density estimation and estimation of regression curves or hazard rates, monotonicity constraints can arise naturally. A frequently encountered problem in nonparametric statistics is to estimate a monotone density function f on a compact interval. A known estimator for density function of f under the restriction that f is decreasing, is Grenander estimator, ...

Random Walks Whose Concave Majorants Often Have Few Faces

We construct a continuous distribution G such that the number of faces in the smallest concave majorant of the random walk with G-distributed summands will take on each natural number infinitely often with probability one. This investigation is motivated by the fact that the number of faces Fn of the concave majorant of the random walk at time n has the same distribution as the number of record...

Contributions to the Theory of Optimal Stopping for One – Dimensional Diffusions

Contributions to the Theory of Optimal Stopping for One–Dimensional Diffusions Savas Dayanik Advisor: Ioannis Karatzas We give a new characterization of excessive functions with respect to arbitrary one–dimensional regular diffusion processes, using the notion of concavity. We show that excessive functions are essentially concave functions, in some generalized sense, and vice–versa. This, in tu...

N ov 2 00 6 A Kiefer Wolfowitz Comparison Theorem For Wicksell ’ s Problem

We extend the isotonic analysis for the Wicksell’s problem to estimate a regression function, which is motivated by the problem of estimating dark matter distribution in Astronomy. The main result is a version of the Kiefer-Wolfowitz theorem comparing the empirical distribution to its least concave majorant, but with a convergence rate n −1 log n faster than n−2/3 log n. The main result is usef...