Related papers: Bi-log-concavity: some properties and some remarks…
We develop an information-theoretic perspective on some questions in convex geometry, providing for instance a new equipartition property for log-concave probability measures, some Gaussian comparison results for log-concave measures, an…
In this article we introduce a three-parameter extension of the bivariate exponential-geometric (BEG) law (Kozubowski and Panorska, 2005). We refer to this new distribution as bivariate gamma-geometric (BGG) law. A bivariate random vector…
We consider the nonparametric regression problem with multiple predictors and an additive error, where the regression function is assumed to be coordinatewise nondecreasing. We propose a Bayesian approach to make an inference on the…
We discuss situations where perturbing a probability measure on $\mathbb{R}^n$ does not deteriorate its Poincar\'e constant by much. A particular example is the symmetric exponential measure in $\mathbb{R}^n$, even log-concave perturbations…
Let $L$ be a countable language. We characterize, in terms of definable closure, those countable theories $\Sigma$ of $\mathcal{L}_{\omega_1, \omega}(L)$ for which there exists an $S_\infty$-invariant probability measure on the collection…
We provide a pointwise bipolar theorem for liminf-closed convex sets of positive Borel measurable functions on a sigma-compact metric space without the assumption that the polar is a tight set of measures. As applications we derive a…
We obtain a formula for the density of the free convolution of an arbitrary probability measure on the unit circle of $\mathbb{C}$ with the free multiplicative analogues of the normal distribution on the unit circle. This description relies…
We introduce a new smooth estimator of the ROC curve based on log-concave density estimates of the constituent distributions. We show that our estimate is asymptotically equivalent to the empirical ROC curve if the underlying densities are…
We construct a transport map from Poisson point processes onto ultra-log-concave measures over the natural numbers, and show that this map is a contraction. Our approach overcomes the known obstacles to transferring functional inequalities…
It is a basic property of the entropy in statistical physics that is concave as a function of energy. The analog of this in representation theory would be the concavity of the logarithm of the multiplicity of an irreducible representation…
We study the linear map sending the numerator of the rational function representing the Hilbert series of a module to that of its r-th Veronese submodule. We show that the asymptotic behaviour as r tends to infinity depends on the…
In this note we study the maximal perimeter of a convex set in $\mathbb{R}^n$ with respect to various classes of measures. Firstly, we show that for a probability measure $\mu$ on $ \mathbb{R}^n$, satisfying very mild assumptions, there…
Diversities have recently been developed as multiway metrics admitting clear and useful notions of hyperconvexity and tight span. In this note we consider the analytic properties of diversities, in particular the generalizations of uniform…
In this paper we study two different weighted isoperimetric inequalities. In the first part of the paper we prove a sharp stability result for the isoperimetric inequality with a log-convex weight. In the second part we analize the behavior…
The bifactor model and its extensions are multidimensional latent variable models, under which each item measures up to one subdimension on top of the primary dimension(s). Despite their wide applications to educational and psychological…
We prove the following type of discrete entropy monotonicity for sums of isotropic, log-concave, independent and identically distributed random vectors $X_1,\dots,X_{n+1}$ on $\mathbb{Z}^d$: $$ H(X_1+\cdots+X_{n+1}) \geq H(X_1+\cdots+X_{n})…
Motivated by studies in biological sciences to detect differentially expressed genes, a semiparametric two-component mixture model with one known component is being studied in this paper. Assuming the density of the unknown component to be…
We propose a log-concavity conjecture for BPS invariants arising in the enumerative geometry of planar curve singularities, identified with the local Euler obstructions of Severi strata in their versal deformations. We further extend this…
Given a probability measure $\mu $ on ${\mathbb R}^n$, Tukey's half-space depth is defined for any $x\in {\mathbb R}^n$ by $\varphi_{\mu }(x)=\inf\{\mu (H):H\in {\cal H}(x)\}$, where ${\cal H}(x)$ is the set of all half-spaces $H$ of…
Let X_1, ..., X_n be independent and identically distributed random vectors with a log-concave (Lebesgue) density f. We first prove that, with probability one, there exists a unique maximum likelihood estimator of f. The use of this…