Fast Algorithms for Nonparametric Curve Estimation

Abstract

Naive implementations of local polynomial fits and kernel estimators require almost O(n ²) operations. In this article two fast O(n) algorithms for nonparametric local polynomial fitting are presented. They are based on updating normal equations. Numerical stability is guaranteed by controlling ill-conditioned situations for small bandwidths and data-tuned restarting of the updating procedure. Restarting at every output point results in a moderately fast but highly stable O(n ^7/5) algorithm. Applicability of algorithms is evaluated for estimation of regression curves and their derivatives. The idea is also applied to kernel estimators of regression curves and densities.     

On conditional extreme values of random vectors with polar representation

An effective approach for studying the asymptotics of bivariate random vectors is to search for the limits of conditional probabilities where the conditioning variable becomes large. In this context, elliptical and related distributions have been extensively investigated. A quite general model was presented by Fougères and Soulier (Limit conditional distributions for bivariate vectors with polar representation in Stochastic Models, 2010), who derived a conditional limit theorem for random vectors (X, Y) with a polar representation R · (u(T), v(T)), where R, T are stochastically independent and R is in the Gumbel max-domain of attraction. We reformulate their assumptions, such that they have a simpler structure, display more clearly the geometry of the curves (u(t), v(t)) and allow us to deduce interesting generalizations into two directions:

u has several global maxima instead of only one,

the curve (u(t), v(t)) is no longer differentiable, but forms a “cusp”.

The latter generalization yields results where only random norming leads to a non-degenerate limit statement. Ideas and results are elucidated by several figures.     

Optimality of the quantified Ingham–Karamata theorem for operator semigroups with general resolvent growth

We prove that a general version of the quantified Ingham–Karamata theorem for $$C_0$$-semigroups is sharp under mild conditions on the resolvent growth, thus generalising the results contained in a recent paper by the same authors. It follows in particular that the well-known Batty–Duyckaerts theorem is optimal even for bounded $$C_0$$-semigroups whose generator has subpolynomial resolvent growth. Our proof is based on an elegant application of the open mapping theorem, which we complement by a crucial technical lemma allowing us to strengthen our earlier results.     

2.2 Organische industrielle Produkte

Ohne Zusammenfassung     

Optimal finite-time processes in stochastic thermodynamics

For a small system like a colloidal particle or a single biomolecule embedded in a heat bath, the optimal protocol of an external control parameter minimizes the mean work required to drive the system from one given equilibrium state to another in a finite time. In general, this optimal protocol obeys an integro-differential equation. Explicit solutions both for a moving laser trap and a time-dependent strength of such a trap show finite jumps of the optimal protocol to be typical both at the beginning and at the end of the process.