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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.

The formation and propagation of thermal fronts in a cylindrical medium that is undergoing microwave heating is studied in detail. The model consists of Maxwell's wave equation coupled to a temperature diffusion equation containing a bistable nonlinear term.

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

Value-at-risk measures the worst loss to be expected of a portfolio over a given time horizon at a given confidence level. Calculation of VaR frequently involves estimating the volatility of return processes and quantiles of standardized returns. In this paper, several semiparametric techniques are introduced to estimate the volatilities. In addition, both parametric and nonparametric techniques are proposed to estimate the quantiles of standardized return processes. The newly proposed techniques also have the flexibility to adapt automatically to the changes in the dynamics of market prices over time. The combination of newly proposed techniques for estimating volatility and standardized quantiles yields several new techniques for evaluating multiple period VaR. The performance of the newly proposed VaR estimators is evaluated and compared with some of existing methods. Our simulation results and empirical