We develop improved statistical procedures for testing the null hypothesis of stochastic monotonicity. Stochastic monotonicity can be reformulated in terms of the concavity of cross-sections of a copula function; our test statistic is based on a empirical measure of departures from concavity. While existing tests of stochastic monotonicity deliver a limiting rejection rate equal to the nominal ...

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...

A new characterization of excessive functions for arbitrary one–dimensional regular diffusion processes is provided, using the notion of concavity. It is shown that excessivity is equivalent to concavity in some suitable generalized sense. This permits a characterization of the value function of the optimal stopping problem as “the smallest nonnegative concave majorant of the reward function” a...

Let (S0, S1, . . . ) be a supermartingale relative to a nondecreasing sequence of σ-algebras H≤0, H≤1, . . . , with S0 ≤ 0 almost surely (a.s.) and differences Xi := Si − Si−1. Suppose that Xi ≤ d and Var(Xi|H≤i−1) ≤ σ 2 i a.s. for every i = 1, 2, . . . , where d > 0 and σi > 0 are non-random constants. Let Tn := Z1 + · · · + Zn, where Z1, . . . , Zn are i.i.d. r.v.’s each taking on only two va...

We extend the isotonic analysis for 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 useful ...

Receiver operating characteristic (ROC) is usually used to analyse the performance of classifiers in data mining. An important ROC analysis topic is ROC convex hull(ROCCH), which is the least convex majorant (LCM) of the empirical ROC curve, and covers potential optima for the given set of classifiers. Generally, ROC performance maximization could be considered to maximize the ROCCH, which also...

A usual technique to generate upper bounds on the optimum of a quadratic 0-1 maximization problem is to consider a linear majorant (LM) of the quadratic objective function f and then solve the corresponding linear relaxation. Several papers have considered LM’s obtained by termwise bounding, but the possibility of bounding groups of terms simultaneously does not appear to have been explored so ...

This paper shows that error bounds can be used as e↵ective tools for deriving complexity results for first-order descent methods in convex minimization. In a first stage, this objective led us to revisit the interplay between error bounds and the KurdykaLojasiewicz (KL) inequality. One can show the equivalence between the two concepts for convex functions having a moderately flat profile near t...

(i.i) ll/IU=( t \ n » — oo makes (L9 l ) into a Banach space. The idea of considering the amalgam (L, l\ as opposed to the Lebesgue space L = (L,l), is a natural one because it allows us to separate the global behavior from the local behavior of a function. This idea goes back to 1926 and Norbert Wiener who considered the special cases (L,1) and (L, l°°) in [Wl] and (L°°, I) and (L\ l°°) in [W2...

‎We define hyperbolic harmonic $omega$-$alpha$-Bloch space‎ ‎$mathcal{B}_omega^alpha$ in the unit ball $mathbb{B}$ of ${mathbb R}^n$ and‎ ‎characterize it in terms of‎ ‎$$frac{omegabig((1-|x|^2)^{beta}(1-|y|^2)^{alpha-beta}big)|f(x)-f(y)|}{[x,y]^gamma|x-y|^{1-gamma}‎},$$ where $0leq gammaleq 1$‎. ‎Similar results are extended to‎ ‎little $omega$-$alpha$-Bloch and Besov spaces‎. ‎These obtained‎...