We discuss the difficulties of estimating quadratic functionals based on observations Y (t) from the white noise model Y (t) where Y (t) is a standard Wiener process on Y (t) . The optimal rates of convergence (as Y (t) ) for estimating quadratic functionals under certain geometric constraints are found. Specifically, the optimal rates of estimating Y (t) under hyperrectangular constraints Y (t) and weighted Y (t) -body constraints Y (t) are computed explicitly, where Y (t) is the Y (t) th Fourier-Bessel coefficient of the unknown function Y (t) . We develop lower bounds based on testing two highly composite hypercubes and address their advantages. The attainable lower bounds are found by applying the hardest one-dimensional approach as well as the hypercube method. We demonstrate that for estimating regular quadratic functionals [ie, the functionals which can be estimated

This paper considers a proportional hazards model, which allows one to examine the extent to which covariates interact nonlinearly with an exposure variable, for analysis of lifetime data. A local partial-likelihood technique is proposed to estimate nonlinear interactions. Asymptotic normality of the proposed estimator is established. The baseline hazard function, the bias and the variance of the local likelihood estimator are consistently estimated. In addition, a one-step local partial-likelihood estimator is presented to facilitate the computation of the proposed procedure and is demonstrated to be as efficient as the fully iterated local partial-likelihood estimator. Furthermore, a penalized local likelihood estimator is proposed to select important risk variables in the model. Numerical examples are used to illustrate the effectiveness of the proposed procedures.

Portfolio allocation with gross-exposure constraint is an effective method to increase the efficiency and stability of portfolios selection among a vast pool of assets, as demonstrated by Fan, Zhang, and Yu. The required high-dimensional volatility matrix can be estimated by using high-frequency financial data. This enables us to better adapt to the local volatilities and local correlations among a vast number of assets and to increase significantly the sample size for estimating the volatility matrix. This article studies the volatility matrix estimation using high-dimensional, high-frequency data from the perspective of portfolio selection. Specifically, we propose the use of pairwise-refresh time and all-refresh time methods based on the concept of refresh time proposed by Barndorff-Nielsen, Hansen, Lunde, and Shephard for the estimation of vast covariance matrix and compare their merits in the portfolio selection. We

Children learn effortlessly by example and exhibit a remarkable capacity of generalization. The field of machine learning, on the other hand, stumbles along clumsily in search of algorithms and methods, but nothing available today comes even close to an average two-year-old toddler. So, modestly, we present some results on one of the main paradigms in machine learningnearest neighbor methods. Rummaging through old data for the closest match seems like a sensible thing to do, and that primitive idea can be formalized and made rigorous. In the field of nonparametric statistics, where one is concerned with the estimation of densities, distribution functions, regression functions, and functionals, the nearest neighbor family of methods was in the limelight from the very beginning and has achieved some level of maturity.

Value at Risk (VaR) is a fundamental tool for managing market risks. It measures the worst loss to be expected of a portfolio over a given time horizon under normal market conditions 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 of the market prices of a portfolio. 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. Their statistical efficiencies are studied both theoretically and empirically. The combination of newly proposed techniques for estimating volatility and standardized quantiles yields several new