The product of two dependent random variables with regularly varying or rapidly varying tails

Let X and Y be two nonnegative and dependent random variables following a generalized Farlie-Gumbel-Morgenstern distribution. In this short note, we study the impact of a dependence structure of X and Y on the tail behavior of XY. We quantify the impact as the limit, as x→∞, of the quotient of Pr(XY>x) and Pr(X^∗Y^∗>x), where X^∗ and Y^∗ are independent random variables identically distributed as X and Y, respectively. We obtain an explicit expression for this limit when X is regularly varying or rapidly varying tailed.     

Stochastic orderings, folded beta distributions and fairness in coin flips

This paper studies stochastic orderings for folded beta distributions. A necessary and sufficient condition for one folded beta random variable to be larger than another in likelihood ratio order is obtained. The work is motivated by the recent results of Porzio and Ragozini on stochastic orderings for folded binomial distributions. Beta distributions are commonly used to model proportions or probabilities, particularly as conjugate distributions for binomial random variables. Stochastic orderings may arise, for instance, in considerations of fairness in coin flips.     

A general criterion to determine the number of change-points

A general criterion is proposed to determine the number K of the change-points in a parametric nonlinear multi-response model. Schwarz criterion is a particular case. The change-points depend on regressor values and not on instant of measure. We prove that the proposed estimator for K is consistent. Simulation results, using Monte Carlo technique, for nonlinear models which have numerous applications, support the relevance of the theory.     

Finite-sample density and its small sample asymptotic approximation

To derive the exact density of a statistic, which can be intractable, is sometimes a difficult problem. The exact densities of estimates of the shift or regression parameters can be derived with the aid of score functions. Moreover, extremely accurate approximations can be obtained by the small sample asymptotics, based on the saddlepoint method. It is of interest to compare these two approaches, at least for small samples. We numerically compare the exact densities of estimates of the shift parameter with their small sample approximations for various parent distributions of the data. For some distributions both methods are in surprising concordance even under very small samples.     

The dispersive effect of cross-aging with archimedean copulas

In this work, we compare conditional distributions derived from bivariate archimedean copulas in terms of their respective variabilities using the dispersive stochastic order. Specifically, we fix the underlying copula and we consider the effect of increasing the second component on the variability of the conditional distribution of the first component. Characterizations are provided in terms of the generator and of the marginal distributions. Several examples involving standard parametric copulas such as Clayton and Frank ones are discussed.     

Commutability of homogenization and linearization at identity in finite elasticity and applications

We prove under some general assumptions on elastic energy densities (namely, frame indifference, minimality at identity, non-degeneracy and existence of a quadratic expansion at identity) that homogenization and linearization commute at identity. This generalizes a recent result by S. Müller and the second author by dropping their assumption of periodicity. As a first application, we extend their Γ-convergence commutation diagram for linearization and homogenization to the stochastic setting under standard growth conditions. As a second application, we prove that the Γ-closure is local at identity for this class of energy densities.     

Symmetry and nonexistence of positive solutions of elliptic equations and systems with Hardy terms

We prove some symmetry property for equations with Hardy terms in cones, without any assumption at infinity. We also show symmetry property and nonexistence of entire solutions of some elliptic systems with Hardy weights.     

Continuous dependence for NLS in fractional order spaces

For the nonlinear Schrödinger equation iut+Δu+λα|u|u=0 in RN, local existence of solutions in Hs is well known in the Hs-subcritical and critical cases 0<α?4/(N−2s), where 0<s<min{N/2,1}. However, even though the solution is constructed by a fixed-point technique, continuous dependence in Hs does not follow from the contraction mapping argument. In this paper, we show that the solution depends continuously on the initial value in the sense that the local flow is continuous Hs→Hs. If, in addition, α?1 then the flow is locally Lipschitz.