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.

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

The varying coefficient model is an important class of nonparametric statistical model, which allows us to examine how the effects of covariates vary with exposure variables. When the number of covariates is large, the issue of variable selection arises. In this article, we propose and investigate marginal nonparametric screening methods to screen variables in sparse ultra-high-dimensional varying coefficient models. The proposed nonparametric independence screening (NIS) selects variables by ranking a measure of the nonparametric marginal contributions of each covariate given the exposure variable. The sure independent screening property is established under some mild technical conditions when the dimensionality is of nonpolynomial order, and the dimensionality reduction of NIS is quantified. To enhance the practical utility and finite sample performance, two data-driven iterative NIS (INIS) methods are