An overview of the goodness-of-fit test problem for copulas

We review the main "omnibus procedures" for goodness-of-fit testing for copulas: tests based on the empirical copula process, on probability integral transformations, on Kendall's dependence function, etc, and some corresponding reductions of dimension techniques. The problems of finding asymptotic distribution-free test statistics and the calculation of reliable p-values are discussed. Some particular cases, like convenient tests for time-dependent copulas, for Archimedean or extreme-value copulas, etc, are dealt with. Finally, the practical performances of the proposed approaches are briefly summarized

Monotone and Consistent discretization of the Monge-Ampere operator

We introduce a novel discretization of the Monge-Ampere operator, simultaneously consistent and degenerate elliptic, hence accurate and robust in applications. These properties are achieved by exploiting the arithmetic structure of the discrete domain, assumed to be a two dimensional cartesian grid. The construction of our scheme is simple, but its analysis relies on original tools seldom encountered in numerical analysis, such as the geometry of two dimensional lattices, and an arithmetic structure called the Stern-Brocot tree. Numerical experiments illustrate the method's efficiency

Diffusion and Correlations in Lattice Gas Automata

We present an analysis of diffusion in terms of the spontaneous density fluctuations in a non-thermal two-species fluid modeled by a lattice gas automaton. The power spectrum of the density correlation function is computed with statistical mechanical methods, analytically in the hydrodynamic limit, and numerically from a Boltzmann expression for shorter time and space scales. In particular we define an observable -- the weighted difference of the species densities -- whose fluctuation correlations yield the diffusive mode independently of the other modes so that the corresponding power spectrum provides a measure of diffusion dynamics solely. Automaton simulations are performed to obtain measurements of the spectral density over the complete range of wavelengths (from the microscopic scale to the macroscopic scale of the automaton universe). Comparison of the theoretical results with the numerical experiments data yields the following results: (i) the spectral functions of the lattice gas fluctuations are in accordance with those of a classical `non-thermal' fluid; (ii) the Landau-Placzek theory, obtained as the hydrodynamic limit of the Boltzmann theory, describes the spectra correctly in the long wavelength limit; (iii) at shorter wavelengths and at moderate densities the complete Boltzmann theory provides good agreement with the simulation data. These results offer convincing validation of lattice gas automata as a microscopic approach to diffusion phenomena in fluid systems.Comment: 9 pages (revtex source), 12 Postscript figure

On kernel-based estimation of conditional Kendall's tau: finite-distance bounds and asymptotic behavior

We study nonparametric estimators of conditional Kendall's tau, a measure of concordance between two random variables given some covariates. We prove non-asymptotic bounds with explicit constants, that hold with high probabilities. We provide "direct proofs" of the consistency and the asymptotic law of conditional Kendall's tau. A simulation study evaluates the numerical performance of such nonparametric estimators.Comment: 29 pages, 4 figures. arXiv admin note: text overlap with arXiv:1802.0761

Single-index copulae

We introduce so-called "single-index copulae". They are semi-parametric conditional copulae whose parameter is an unknown "link" function of a univariate index only. We provide estimates of this link function and of the finite dimensional unknown parameter. The asymptotic properties of the latter estimates are stated. Thanks to some properties of conditional Kendall's tau, we illustrate our technical conditions with several usual copula families.Comment: Revised version: correction of Assumption 3 and some minor induced modification