In this paper, we provide the O() corrections to the hydrodynamic model derived by Degond and Motsch from a kinetic version of the model by Vicsek and co-authors describing flocking biological agents. The parameter stands for the ratio of the microscopic to the macroscopic scales. The O() corrected model involves diffusion terms in both the mass and velocity equations as well as terms which are quadratic functions of the first-order derivatives of the density and velocity. The derivation method is based on the standard ChapmanEnskog theory, but is significantly more complex than usual due to both the non-isotropy of the fluid and the lack of momentum conservation.

Generalized linear models and quasi-likelihood method extend the ordinary regression models to accommodate more general conditional distributions of the response. Nonparametric methods need no explicit parametric specification and the resulting model is completely determined by the data themselves. However nonparametric estimation schemes generally have a slower convergence rate such as the local polynomial smoothing estimation of nonparametric generalized linear models studied in Fan, Heckman and Wand (1995). In this work, we propose two parametrically guided nonparametric estimation schemes by incorporating prior shape information on the link transformation of the response variables conditional mean in terms of the predictor variable. Asymptotic results and numerical simulations demonstrate the improvement of our new estimation schemes over the original nonparametric counterpart.

When estimating a mean regression function and its derivatives, locally weighted least squares regression has proven to be a very attractive technique. The present paper focuses on the important issue of how to select the smoothing parameter or bandwidth. In the case of estimating curves with a complicated structure, a variable bandwidth is desirable. Furthermore, the bandwidth should be indicated by the data themselves. Recent developments in nonparametric smoothing techniques inspired us to propose such a datadriven bandwidth selection procedure, which can be used to select both constant and variable bandwidths. The idea is based on a residual squares criterion along with a good approximation of the bias and variance of the estimator. The procedure can be applied to select bandwidths not only for estimating the regression curve but also for estimating its derivatives. The resulting estimation

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