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

The paper studies estimation of partially linear hazard regression models with varying coefficients for multivariate survival data. A profile pseudopartiallikelihood estimation method is proposed. The estimation of the parameters of the linear part is accomplished via maximization of the profile pseudopartiallikelihood, whereas the varyingcoefficient functions are considered as nuisance parameters that are profiled out of the likelihood. It is shown that the estimators of the parameters are root <i>n</i> consistent and the estimators of the nonparametric coefficient functions achieve optimal convergence rates. Asymptotic normality is obtained for the estimators of the finite parameters and varyingcoefficient functions. Consistent estimators of the asymptotic variances are derived and empirically tested, which facilitate inference for the model. We prove that the varyingcoefficient functions can be estimated as well as if the

Let E be a real Banach space with a uniformly Gteaux differentiable norm and which possesses uniform normal structure, K a nonempty bounded closed convex subset of E,{T i} i= 1 N a finite family of asymptotically nonexpansive self-mappings on K with common sequence {k n} n= 1[1,),{t n},{s n} be two sequences in (0, 1) such that s n+ t n= 1 (n 1) and f be a contraction on K. Under suitable conditions on the sequences {s n},{t n}, we show the existence of a sequence {x n} satisfying the relation x n=(1 1 k n) x n+ s n k n f (x n)+ t n k n T r n n x n where n= l n N+ r n for some unique integers l n 0 and 1 r n N. Further we prove that {x n} converges strongly to a common fixed point of {T i} i= 1 N, which solves some variational inequality, provided x n T i x n 0 as n for i= 1, 2,, N. As an application, we prove that the iterative process defined by z 0 K, z n+ 1=(1 1 k n) z n+ s n k n f (z n)+ t n k n

Schrodinger operators with strongly singular interactions, typically supported by a set of Lebesgue measure zero, eg, by a nite or countable set of points, appeared in quantum mechanics from the very beginning~cf., eg,[35, 28]. Their use had essentially two motivations. First of all, they were expected to provide a reasonable description in the physical situations when the real interaction has a very small range comparing to the other characteristic lengths of the problem, for instance, to the wavelength of the scattered particles. On the other hand, the fact that the interaction support is small and outside it a free solution of the corresponding Schrodinger equation may be used often makes such models explicitly solvable. Until the beginning of the sixties, however, the occasional use of these point interactions remained purely formal. Only after the work of Berezin and Faddeev [11] did it become clear they can be

Consider an unstable quantum system that has been found undecayed at an instant s and denote by R(t,s) the rate of its regeneration into an original (undecayed) state at a later instant t+s. It is proved that the reduced evolution is a semigroup, i.e., there is no regeneration at all, provided R(t,s)1/2 can be estimated by a sufficiently regular function that is nondecreasing in s and has zero derivative with respect to t at t=0 for every s. This generalizes the theorem of Misra and Sinha [Helv. Phys. Acta 45, 619 (1972)] in a different direction than in a recent paper by Nishioka [J. Math. Phys. 29, 1860 (1988)].