Asymmetric vector moving average models: estimation and testing

We propose the class of asymmetric vector moving average (asVMA) models. The asymmetry of these models is characterized by different MA filters applied to the components of vectors of lagged positive and negative innovations. This allows for a detailed investigation of the interrelationships among past model innovations of different sign. We derive some covariance matrix properties of an asVMA model under the assumption of Gaussianity. Related to this, we investigate the global invertibility condition of the proposed model. The paper also introduces a maximum likelihood estimation procedure and a multivariate Wald-type test statistic for symmetry versus the alternative of asymmetry. The finite-sample performance of the proposed multivariate test is studied by simulation. Furthermore, we devise an exploratory test statistic based on lagged sample cross-bicovariance estimates. The estimation and testing procedures are used to uncover asymmetric effects in two US growth rates, and in three US industrial prices.

On solvability of elliptic boundary value problems via global invertibility

In this work we apply global invertibility result in order to examine the solvability of elliptic equations with both Neumann and Dirichlet boundary conditions.

Global invertibility of Sobolev maps

We define a class of Sobolev {W^{1,p}(\Omega,\mathbb{R}^{n})} functions, with {p>n-1} , such that its trace on {\partial\Omega} is also Sobolev, and do not present cavitation in the interior or on the boundary. We show that if a function in this class has positive Jacobian and coincides on the boundary with an injective map, then the function is itself injective. We then prove the existence of minimizers within this class for the type of functionals that appear in nonlinear elasticity.

Global injectivity in second-gradient nonlinear elasticity and its approximation with penalty terms

We present a new penalty term approximating the Ciarlet–Nečas condition (global invertibility of deformations) as a soft constraint for hyperelastic materials. For non-simple materials including a suitable higher-order term in the elastic energy, we prove that the penalized functionals converge to the original functional subject to the Ciarlet–Nečas condition. Moreover, the penalization can be chosen in such a way that for all low-energy deformations, self-interpenetration is avoided completely already at all sufficiently small finite values of the penalization parameter. We also present numerical experiments in two dimensions illustrating our theoretical results and provide own MATLAB code available for download and testing.