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

Factor modeling is an essential tool for exploring intrinsic dependence structures among high-dimensional random variables. Much progress has been made for estimating the covariance matrix from a high-dimensional factor model. However, the blessing of dimensionality has not yet been fully embraced in the literature: much of the available data are often ignored in constructing covariance matrix estimates. If our goal is to accurately estimate a covariance matrix of a set of targeted variables, shall we employ additional data, which are beyond the variables of interest, in the estimation? In this article, we provide sufficient conditions for an affirmative answer, and further quantify its gain in terms of Fisher information and convergence rate. In fact, even an oracle-like result (as if all the factors were known) can be achieved when a sufficiently large number of variables is used. The idea of using data as much as possible

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Error variance estimation plays an important role in statistical inference for high-dimensional regression models. This article concerns with error variance estimation in high-dimensional sparse additive model. We study the asymptotic behavior of the traditional mean squared errors, the naive estimate of error variance, and show that it may significantly underestimate the error variance due to spurious correlations that are even higher in nonparametric models than linear models. We further propose an accurate estimate for error variance in ultrahigh-dimensional sparse additive model by effectively integrating sure independence screening and refitted cross-validation techniques. The root <i>n</i> consistency and the asymptotic normality of the resulting estimate are established. We conduct Monte Carlo simulation study to examine the finite sample performance of the newly proposed estimate. A real data example is used to

We study an accelerated Langevin equation in which the classical overdamped Langevin drift is supplemented by a nonlocal gradient velocity determined through a target-weighted Poisson equation. The target \(p\) is only assumed to be strictly positive and need not be normalized. Writing the dynamics in terms of \(\omega=\log(p/q)\) yields a coupled parabolic--elliptic system whose parabolic component contains the usual Langevin diffusion, whereas the elliptic component generates a global transport. The system is invariant under multiplication of \(p\) by a positive constant. On a bounded smooth domain with homogeneous Neumann boundary conditions, we establish global existence and uniqueness of classical solutions for strictly positive \(C^{2+\alpha}\) target weights and compatible \(C^{2+\alpha}\) initial data, with full \(\alpha\)-H\"older regularity up to the initial time. For long-time behavior, the oscillation of \(\omega\) decays at the explicit rate \(e^{-t}\). For normalized initial densities, the solution density converges exponentially to the normalized target. Moreover, an exact entropy-dissipation identity holds relative to the normalized target, in which the relative Fisher information and the Jeffreys divergence give the two dissipation mechanisms associated with the Langevin and weighted-Poisson components. In particular, KL divergence also converges to zero with rate $e^{-t}$ without a convexity assumption on the target.