Propagation of Gevrey regularity for solutions of Landau equations

By using the energy-type inequality, we obtain, in this paper, the result on propagation of Gevrey regularity for the solution of the spatially homogeneous Landau equation in the cases of Maxwellian molecules and hard potential

Gevrey Regularity for Solution of the Spatially Homogeneous Landau Equation

In this paper, we study the Gevrey class regularity for solutions of the spatially homogeneous Landau equations in the hard potential case and the Maxwellian molecules case

Analytic smoothness effect of solutions for spatially homogeneous Landau equation

In this paper, we study the smoothness effect of Cauchy problem for the spatially homogeneous Landau equation in the hard potential case and the Maxwellian molecules case. We obtain the analytic smoothing effect for the solutions under rather weak assumptions on the initial datum.Comment: 16 page

Semiparametric GEE analysis in partially linear single-index models for longitudinal data

In this article, we study a partially linear single-index model for longitudinal data under a general framework which includes both the sparse and dense longitudinal data cases. A semiparametric estimation method based on a combination of the local linear smoothing and generalized estimation equations (GEE) is introduced to estimate the two parameter vectors as well as the unknown link function. Under some mild conditions, we derive the asymptotic properties of the proposed parametric and nonparametric estimators in different scenarios, from which we find that the convergence rates and asymptotic variances of the proposed estimators for sparse longitudinal data would be substantially different from those for dense longitudinal data. We also discuss the estimation of the covariance (or weight) matrices involved in the semiparametric GEE method. Furthermore, we provide some numerical studies including Monte Carlo simulation and an empirical application to illustrate our methodology and theory.Comment: Published at http://dx.doi.org/10.1214/15-AOS1320 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org