Quadratic regression functionals are important for bandwidth selection of nonparametric regression techniques and for nonparametric goodness-of-fit test. Based on local polynomial regression, we propose estimators for weighted integrals of squared derivatives of regression functions. The rates of convergence in mean square error are calculated under various degrees of smoothness and appropriate values of the smoothing parameter. Asymptotic distributions of the proposed quadratic estimators are considered with the Gaussian noise assumption. It is shown that when the estimators are pseudo-quadratic (linear components dominate quadratic components), asymptotic normality with rate n-1/2 can be achieved.

Generalized linear models and quasi-likelihood method extend the ordinary regression models to accommodate more general conditional distributions of the response. Nonparametric methods need no explicit parametric specification and the resulting model is completely determined by the data themselves. However nonparametric estimation schemes generally have a slower convergence rate such as the local polynomial smoothing estimation of nonparametric generalized linear models studied in Fan, Heckman and Wand (1995). In this work, we propose two parametrically guided nonparametric estimation schemes by incorporating prior shape information on the link transformation of the response variables conditional mean in terms of the predictor variable. Asymptotic results and numerical simulations demonstrate the improvement of our new estimation schemes over the original nonparametric counterpart.

In statistics and machine learning, people are often interested in the eigenvectors (or singular vectors) of certain matrices (eg covariance matrices, data matrices, etc). However, those matrices are usually perturbed by noises or statistical errors, either from random sampling or structural patterns. One usually employs Davis-Kahan \sin theorem to bound the difference between the eigenvectors of a matrix \sin and those of a perturbed matrix \sin , in terms of \sin norm. In this paper, we prove that when \sin is a low-rank and incoherent matrix, the \sin norm perturbation bound of singular vectors (or eigenvectors in the symmetric case) is smaller by a factor of \sin or \sin for left and right vectors, where \sin and \sin are the matrix dimensions. The power of this new perturbation result is shown in robust covariance estimation, particularly when random variables have heavy tails. There, we propose new robust covariance estimators and establish their asymptotic properties using the newly developed perturbation bound. Our theoretical results are verified through extensive numerical experiments.

Parametric option pricing models are widely used in finance. These models capture several features of asset price dynamics; however, their pricing performance can be significantly enhanced when they are combined with nonparametric learning approaches that learn and correct empirically the pricing errors. In this article we propose a new nonparametric method for pricing derivatives assets. Our method relies on the state price distribution instead of the state price density, because the former is easier to estimate nonparametrically than the latter. A parametric model is used as an initial estimate of the state price distribution. Then the pricing errors induced by the parametric model are fitted nonparametrically. This model-guided method, called automatic correction of errors (ACE), estimates the state price distribution nonparametrically. The method is easy to implement and can be combined with any model-based

Long term prediction such as multi-step time series prediction is a challenging prognostics problem. This paper proposes an improved AR time series model called ND-AR model (Nonlinear Degradation AutoRegression) for Remaining Useful Life (RUL) estimation of lithium-ion batteries. The nonlinear degradation feature of the lithiumion battery capacity degradation is analyzed and then the non-linear accelerated degradation factor is extracted to improve the linear AR model. In this model, the nonlinear degradation factor can be obtained with curve fitting, and then the ND-AR model can be applied as an adaptive datadriven prognostics method to monitor degradation time series data. Experimental results with CALCE battery data set show that the proposed nonlinear degradation AR model can realize satisfied prognostics for various lithium-ion batteries with low computing complexity.