We propose a high-dimensional classification method that involves nonparametric feature augmentation. Knowing that marginal density ratios are the most powerful univariate classifiers, we use the ratio estimates to transform the original feature measurements. Subsequently, penalized logistic regression is invoked, taking as input the newly transformed or augmented features. This procedure trains models equipped with local complexity and global simplicity, thereby avoiding the curse of dimensionality while creating a flexible nonlinear decision boundary. The resulting method is called feature augmentation via nonparametrics and selection (FANS). We motivate FANS by generalizing the naive Bayes model, writing the log ratio of joint densities as a linear combination of those of marginal densities. It is related to generalized additive models, but has better interpretability and computability. Risk bounds are

This article proposes a consistent and efficient estimator of the high-frequency covariance (quadratic covariation) of two arbitrary assets, observed asynchronously with market microstructure noise. This estimator is built on the marriage of the quasimaximum likelihood estimator of the quadratic variation and the proposed generalized synchronization scheme and thus is not influenced by the Epps effect. Moreover, the estimation procedure is free of tuning parameters or bandwidths and is readily implementable. Monte Carlo simulations show the advantage of this estimator by comparing it with a variety of estimators with specific synchronization methods. The empirical studies of six foreign exchange future contracts illustrate the time-varying correlations of the currencies during the 2008 global financial crisis, demonstrating the similarities and differences in their roles as key currencies in the global market.

We study a system of semilinear hyperbolic equations passively advected by smooth white noise in time random velocity fields. Such a system arises in modelling non-premixed isothermal turbulent flames under single-step kinetics of fuel and oxidizer. We derive closed equations for one-point and multi-point probability distribution functions (PDFs) and closed-form analytical formulae for the one-point PDF function, as well as the two-point PDF function under homogeneity and isotropy. Exact solution formulae allow us to analyse the ensemble-averaged fuel/oxidizer concentrations and the motion of their level curves. We recover the empirical formulae of combustion in the thin reaction zone limit and show that these approximate formulae can either underestimate or overestimate average concentrations when the reaction zone is not tending to zero. We show that the averaged reaction rate slows down locally in

An effective bandwidth selection method for local linear regression is proposed in Fan and Gijbels [1995, J. Roy. Statist. Soc. Ser. B, 57, 371394]. The method is based on the idea of the pre-asymptotic substitution and has been tested extensively. This paper investigates the rate of convergence of this method. In particular, we show that the relative rate of convergence is of order <i>n</i>^2/7 if the locally cubic fitting is used in the pilot stage, and the rate of convergence is <i>n</i>^2/5 when the local polynomial of degree 5 is used in the pilot fitting. The study also reveals a marked difference between the bandwidth selection for nonparametric regression and that for density estimation: The plug-in approach for the latter case can admit the root-<i>n</i> rate of convergence while for the former case the best rate is of order <i>n</i>^2/5.

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