We prove that a positive function on the unit disk admits a harmonic majorant if and only if a certain explicit upper envelope of it admits a superharmonic majorant. We provide examples to show that mere superharmonicity of the data does not help with the problem of existence of a harmonic majorant. 1. Definitions and statements Let D stand for the open unit disk in the complex plane, and H(D) ...

This article proposes an omnibus test for monotonicity of nonparametric conditional distributions and its moments. Unlike previous proposals, our method does not require smooth estimation of the derivatives of nonparametric curves and it can be implemented even when probability densities do not exist. In fact, we only require continuity of the marginal distributions under the null and fixed alt...

We verify functional a posteriori error estimates proposed by S. Repin for a class of obstacle problems in two space dimensions. New benchmarks with known analytical solution are constructed based on one dimensional benchmark introduced by P. Harasim and J. Valdman. Numerical approximation of the solution of the obstacle problem is obtained by the finite element method using bilinear elements o...

Two probability distributions with common support are said to exhibit density ratio ordering when they admit a nonincreasing density ratio. Existing statistical tests of the null hypothesis of density ratio ordering are known to be conservative, with null limiting rejection rates below the nominal significance level whenever the two distributions are unequal. We show how a bootstrap procedure c...

In this paper we deal with a theoretical question raised in connection with the application of Dirichlet-Sch type inequality, obtained by Huang (Int. Math. J. 27(02):1650009, 2016), which has been already applied to obtain multiplicity results for boundary value problems in several recent papers. We also discuss a particular case of it in more detail. As an application, we deduce the least harm...

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 ...

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...