We prove a sharp Bernstein inequality for general-state-space and not necessarily reversible Markov chains. It is sharp in the sense that the variance proxy term is optimal. Our result covers the classical Bernstein's inequality for independent random variables as a special case.

Suppose we lhave n observationis ftom Y= X+ c, where e is nieasuremiienit error with kiowin distributioni, and the denesity f of X is uniknowii withi thte nloln-paraniietric conistrainit

No abstract uploaded!

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

Error variance estimation plays an important role in statistical inference for high-dimensional regression models. This article concerns with error variance estimation in high-dimensional sparse additive model. We study the asymptotic behavior of the traditional mean squared errors, the naive estimate of error variance, and show that it may significantly underestimate the error variance due to spurious correlations that are even higher in nonparametric models than linear models. We further propose an accurate estimate for error variance in ultrahigh-dimensional sparse additive model by effectively integrating sure independence screening and refitted cross-validation techniques. The root <i>n</i> consistency and the asymptotic normality of the resulting estimate are established. We conduct Monte Carlo simulation study to examine the finite sample performance of the newly proposed estimate. A real data example is used to