An application of the convolution inequality for the Fisher information

A characterization of the normal distribution by a statistical independence on a linear transformation of two mutually independent random variables is proved by using the convolution inequality for the Fisher information.     

Eigenvalue gaps for the Cauchy process and a Poincaré inequality

A connection between the semigroup of the Cauchy process killed upon exiting a domain D and a mixed boundary value problem for the Laplacian in one dimension higher known as the mixed Steklov problem, was established in [R. Bañuelos, T. Kulczycki, The Cauchy process and the Steklov problem, J. Funct. Anal. 211 (2004) 355-423]. From this, a variational characterization for the eigenvalues λn, n?1, of the Cauchy process in D was obtained. In this paper we obtain a variational characterization of the difference between λn and λ₁. We study bounded convex domains which are symmetric with respect to one of the coordinate axis and obtain lower bound estimates for λ_∗−λ₁ where λ_∗ is the eigenvalue corresponding to the “first” antisymmetric eigenfunction for D. The proof is based on a variational characterization of λ_∗−λ₁ and on a weighted Poincaré-type inequality. The Poincaré inequality is valid for all α symmetric stable processes, 0<α?2, and any other process obtained from Brownian motion by subordination. We also prove upper bound estimates for the spectral gap λ₂−λ₁ in bounded convex domains.     

Efficiency and Cramér-Rao type lnequalities for convex loss functions

A general method for obtaining inequalities of Cramér-Rao type for convex loss functions is presented. It is shown under rather weak assumptions that there are at least as many such inequalities as best unbiased estimators. More precisely, it is shown that an estimator is efficient with respect to an inequality of Cramér-Rao type if and only if it is the best in the class of unbiased estimators. Moreover, theorems of Blyth and Roberts (“Proceedings Sixth Berkeley Symposium on Math. Statist. Prob.,”) and of Blyth (Ann. Statist.2, 464–473) are extended. We make an use of methods of convex analysis and properties of convex integral functionals on Orlicz spaces.     

Is an Orlicz–Poincaré inequality an open ended condition,and what does that mean?

In [16], Keith and Zhong prove that spaces admitting Poincaré inequalities also admit a priori stronger Poincaré inequalities. We use their technique, with slight adjustments, to obtain a similar result in the case of Orlicz–Poincaré inequalities. We give examples in the plane that show all hypotheses are required.     

On information inequalities in sequential estimation for stochastic processes

Information inequalities in a general sequential model for stochastic processes are presented by applying the approach to estimation through estimating functions. Using this approach, Bayesian versions of the information inequalities are also obtained. In particular, exponential-family processes and counting processes are considered. The results are useful to find optimum properties of parameter estimators. The assertions are of great importance for describing estimators in failure-repair models in both Bayes approach and the nuisance parameter case.     

Heinz‐type inequality and bi‐Lipschitz continuity for quasiconformal mappings satisfying inhomogeneous biharmonic equations

Let be the class of all sense‐preserving homeomorphic self‐mappings of . The aim of this paper is twofold. First, we obtain Heinz‐type inequality for (K,K′)‐quasiconformal mappings satisfying inhomogeneous biharmonic equation Δ(Δω) = g in unit disk with associated boundary value conditions and . Second, we establish biLipschitz continuity for (K,K′)‐quasiconformal mappings satisfying aforementioned inhomogeneous biharmonic equation when and are small enough.     

Multivariate time changes for Lévy asset models: Characterization and calibration

We build a theoretical framework for multivariate subordination of Brownian motions, with a common and an idiosyncratic component. This follows economic intuition and introduces generalizations of some well known multivariate Lévy processes for financial applications: the compound Poisson, NIG, Variance Gamma and CGMY. In most cases we obtain the characteristic function in closed form. The extension is first kept parsimonious, by adding one parameter only. The empirical fit of (linear) dependence is then increased, by allowing for dependent Brownian motions.     

Local smoothing property and Strichartz inequality for Schrödinger equations with potentials superquadratic at infinity

We study smoothing properties for time-dependent Schrödinger equations , , with potentials which satisfy V(x)=O(|x|^m) at infinity, m?2. We show that the solution u(t,x) is 1/m times differentiable with respect to x at almost all , and explain that this is the result of the fact that the sojourn time of classical particles with energy λ in arbitrary compact set is less than CTλ^−1/m during [0,T] when λ is very large. We also show Strichartz's inequality with derivative loss for such potentials and give its application to nonlinear Schrödinger equations.     

Interior continuity,continuity up to the boundary,and Harnack's inequality for double-phase elliptic equations with nonlogarithmic conditions

We prove continuity and Harnack's inequality for bounded solutions to elliptic equations of the type under the precise choice of μ.      

(h, Φ)-entropy differential metric

Burbea and Rao (1982a, 1982b) gave some general methods for constructing quadratic differential metrics on probability spaces. Using these methods, they obtained the Fisher information metric as a particular case. In this paper we apply the method based on entropy measures to obtain a Riemannian metric based on (h, )-entropy measures (Salicrú et al., 1993). The geodesic distances based on that information metric have been computed for a number of parametric families of distributions. The use of geodesic distances in testing statistical hypotheses is illustrated by an example within the Pareto family. We obtain the asymptotic distribution of the information matrices associated with the metric when the parameter is replaced by its maximum likelihood estimator. The relation between the information matrices and the Cramér-Rao inequality is also obtained.     

General auxiliary problem and algorithm for a generalized multi-valued variational-like inequality problem in reflexive Banach spaces

In this article, some generalized multi-valued variational-like inequality problems and their related general auxiliary problems in real reflexive Banach spaces are introduced. An existence theorem for one of the general auxiliary problems is proved. Furthermore, by exploiting this theorem, an algorithm for the corresponding generalized multi-valued variational-like inequality problem is constructed. The theorems of this article generalize, improve and unify many known corresponding results.     

Poincaré inequalities, uniform domains and extension properties for Newton-Sobolev functions in metric spaces

In the setting of metric measure spaces equipped with a doubling measure supporting a weak p-Poincaré inequality with 1?p<∞, we show that any uniform domain Ω is an extension domain for the Newtonian space N1,p(Ω) and that Ω, together with the metric and the measure inherited from X, supports a weak p-Poincaré inequality. For p>1, we obtain a near characterization of N1,p-extension domains with local estimates for the extension operator.