For the data based choice of the bandwidth of a kernel density estimator, several methods have recently been proposed which have a very fast asymptotic rate of convergence to the optimal bandwidth. In particular the relative rate of convergence is the square root of the sample size, which is known to be the best possible. The point of this paper is to show how semiparametric arguments can be employed to calculate the best possible constant coefficient, that is, an analog of the usual Fisher information, in this convergence. This establishes an important benchmark as to how well bandwidth selection methods can ever hope to perform. It is seen that some existing methods attain the bound, others do not.

Semiparametric regression models are very useful for longitudinal data analysis. The complexity of semiparametric models and the structure of longitudinal data pose new challenges to parametric inferences and model selection that frequently arise from longitudinal data analysis. In this article, two new approaches are proposed for estimating the regression coefficients in a semiparametric model. The asymptotic normality of the resulting estimators is established. An innovative class of variable selection procedures is proposed to select significant variables in the semiparametric models. The proposed procedures are distinguished from others in that they simultaneously select significant variables and estimate unknown parameters. Rates of convergence of the resulting estimators are established. With a proper choice of regularization parameters and penalty functions, the proposed variable selection procedures

In the analysis of cluster data the regression coefficients are frequently assumed to be the same across all clusters. This hampers the ability to study the varying impacts of factors on each cluster. In this paper, a semiparametric model is introduced to account for varying impacts of factors over clusters by using cluster-level covariates. It achieves the parsimony of parametrization and allows the explorations of nonlinear interactions. The random effect in the semiparametric model accounts also for within cluster correlation. Local linear based estimation procedure is proposed for estimating functional coefficients, residual variance, and within cluster correlation matrix. The asymptotic properties of the proposed estimators are established and the method for constructing simultaneous confidence bands are proposed and studied. In addition, relevant hypothesis testing problems are addressed. Simulation studies are

No abstract uploaded!

We present a novel variation of the well-known infomax algorithm of blind source separation. Under natural gradient descent, the infomax algorithm converges to a stationary point of a limiting ordinary differential equation. However, due to the presence of saddle points or local minima of the corresponding likelihood function, the algorithm may be trapped around these bad stationary points for a long time, especially if the initial data are near them. To speed up convergence, we propose to add a sequence of random perturbations to the infomax algorithm to shake the iterating sequence so that it is captured by a path descending to a more stable stationary point. We analyze the convergence of the randomly perturbed algorithm, and illustrate its fast convergence through numerical examples on blind demixing of stochastic signals. The examples have analytical structures so that saddle points or local minima of the likelihood functions are explicit. The results may have implications for online learning algorithms in dissimilar problems.