This paper introduces a Projected Principal Component Analysis (Projected-PCA), which employees principal component analysis to the projected (smoothed) data matrix onto a given linear space spanned by covariates. When it applies to high-dimensional factor analysis, the projection removes noise components. We show that the unobserved latent factors can be more accurately estimated than the conventional PCA if the projection is genuine, or more precisely, when the factor loading matrices are related to the projected linear space. When the dimensionality is large, the factors can be estimated accurately even when the sample size is finite. We propose a flexible semi-parametric factor model, which decomposes the factor loading matrix into the component that can be explained by subject-specific covariates and the orthogonal residual component. The covariates effects on the factor loadings are further

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

When estimating a mean regression function and its derivatives, locally weighted least squares regression has proven to be a very attractive technique. The present paper focuses on the important issue of how to select the smoothing parameter or bandwidth. In the case of estimating curves with a complicated structure, a variable bandwidth is desirable. Furthermore, the bandwidth should be indicated by the data themselves. Recent developments in nonparametric smoothing techniques inspired us to propose such a datadriven bandwidth selection procedure, which can be used to select both constant and variable bandwidths. The idea is based on a residual squares criterion along with a good approximation of the bias and variance of the estimator. The procedure can be applied to select bandwidths not only for estimating the regression curve but also for estimating its derivatives. The resulting estimation

<b></b> Estimation of the covariance structure of longitudinal processes is a fundamental prerequisite for the practical deployment of functional mapping designed to study the genetic regulation and network of quantitative variation in dynamic complex traits. We present a nonparametric approach for estimating the covariance structure of a quantitative trait measured repeatedly at a series of time points. Specifically, we adopt Huang et al.'s (2006,<i>Biometrika</i><b>93</b>, 8598) approach of invoking the modified Cholesky decomposition and converting the problem into modeling a sequence of regressions of responses. A regularized covariance estimator is obtained using a normal penalized likelihood with an<i>L</i>₂ penalty. This approach, embedded within a mixture likelihood framework, leads to enhanced accuracy, precision, and flexibility of functional mapping while preserving its biological relevance. Simulation studies are

We develop a specification test for the transition density of a discretely sampled continuous-time jump-diffusion process, based on a comparison of a nonparametric estimate of the transition density or distribution function with their corresponding parametric counterparts assumed by the null hypothesis. As a special case, our method applies to pure diffusions. We provide a direct comparison of the two densities for an arbitrary specification of the null parametric model using three different discrepancy measures between the null and alternative transition density and distribution functions. We establish the asymptotic null distributions of proposed test statistics and compute their power functions. We investigate the finite-sample properties through simulations and compare them with those of other tests. This article has supplementary material online.

Most papers on high-dimensional statistics are based on the assumption that none of the regressors are correlated with the regression error, namely, they are exogeneous. Yet, endogeneity arises easily in high-dimensional regression due to a large pool of regressors and this causes the inconsistency of the penalized least-squares methods and possible false scientific discoveries. A necessary condition for model selection of a very general class of penalized regression methods is given, which allows us to prove formally the inconsistency claim. To cope with the possible endogeneity, we construct a novel penalized focussed generalized method of moments (FGMM) criterion function and offer a new optimization algorithm. The FGMM is not a smooth function. To establish its asymptotic properties, we first study the model selection consistency and an oracle property for a general class of penalized regression methods. These results are then used to show that the FGMM possesses an oracle property even in the presence of endogenous predictors, and that the solution is also near global minimum under the over-identification assumption. Finally, we also show how the semi-parametric efficiency of estimation can be achieved via a two-step approach.

Statistical and machine learning theory has developed several conditions ensuring that popular estimators such as the Lasso or the Dantzig selector perform well in high-dimensional sparse regression, including the restricted eigenvalue, compatibility, and q sensitivity properties. However, some of the central aspects of these conditions are not well understood. For instance, it is unknown if these conditions can be checked efficiently on any given dataset. This is problematic, because they are at the core of the theory of sparse regression. Here we provide a rigorous proof that these conditions are NP-hard to check. This shows that the conditions are computationally infeasible to verify, and raises some questions about their practical applications. However, by taking an average-case perspective instead of the worst-case view of NP-hardness, we show that a particular condition, q sensitivity, has certain

Two measures of sensitivity to initial conditions in nonlinear time series are proposed. The notions give some insight into the relationship between the Fisher information in statistical estimation and initial-value sensitivity in dynamical systems. By using the locally polynomial regression, we develop nonparametric estimates for a conditional density function, its square root and its partial derivatives. The proposed procedures are innovative and of interests in their own right. They are also used to estimate the sensitive measures. The asymptotic normality of the proposed estimators have been proved. We also propose a simple and intuitively appealing method for choosing the bandwidths. Two simulated examples are used as illustrations.

Varyingcoefficient linear models arise from multivariate nonparametric regression, nonlinear time series modelling and forecasting, functional data analysis, longitudinal data analysis and others. It has been a common practice to assume that the varying coefficients are functions of a given variable, which is often called an <i>index</i>. To enlarge the modelling capacity substantially, this paper explores a class of varyingcoefficient linear models in which the index is unknown and is estimated as a linear combination of regressors and/or other variables. We search for the index such that the derived varyingcoefficient model provides the least squares approximation to the underlying unknown multidimensional regression function. The search is implemented through a newly proposed hybrid backfitting algorithm. The core of the algorithm is the alternating iteration between estimating the index through a onestep scheme

We examine the effect of errors in covariates in nonparametric function estimation. These functions include densities, regressions and conditional quantiles. To estimate these functions, we use the idea of deconvoluting kernels in conjunction with the ordinary kernel methods. We also discuss a new class of function estimators based on local polynomials.