Factor modeling is an essential tool for exploring intrinsic dependence structures among high-dimensional random variables. Much progress has been made for estimating the covariance matrix from a high-dimensional factor model. However, the blessing of dimensionality has not yet been fully embraced in the literature: much of the available data are often ignored in constructing covariance matrix estimates. If our goal is to accurately estimate a covariance matrix of a set of targeted variables, shall we employ additional data, which are beyond the variables of interest, in the estimation? In this article, we provide sufficient conditions for an affirmative answer, and further quantify its gain in terms of Fisher information and convergence rate. In fact, even an oracle-like result (as if all the factors were known) can be achieved when a sufficiently large number of variables is used. The idea of using data as much as possible

Two measures of sensitivity to initial conditions in nonlinear time series are proposed. The notions give some insight into the relationship between the Fisher information in statistical estimation and initial-value sensitivity in dynamical systems. By using the locally polynomial regression, we develop nonparametric estimates for a conditional density function, its square root and its partial derivatives. The proposed procedures are innovative and of interests in their own right. They are also used to estimate the sensitive measures. The asymptotic normality of the proposed estimators have been proved. We also propose a simple and intuitively appealing method for choosing the bandwidths. Two simulated examples are used as illustrations.

Varyingcoefficient linear models arise from multivariate nonparametric regression, nonlinear time series modelling and forecasting, functional data analysis, longitudinal data analysis and others. It has been a common practice to assume that the varying coefficients are functions of a given variable, which is often called an <i>index</i>. To enlarge the modelling capacity substantially, this paper explores a class of varyingcoefficient linear models in which the index is unknown and is estimated as a linear combination of regressors and/or other variables. We search for the index such that the derived varyingcoefficient model provides the least squares approximation to the underlying unknown multidimensional regression function. The search is implemented through a newly proposed hybrid backfitting algorithm. The core of the algorithm is the alternating iteration between estimating the index through a onestep scheme