Averaged Singular Integral Estimation as a Bias Reduction Technique

This paper proposes an averaged version of singular integral estimators, whose bias achieves higher rates of convergence under smoothing assumptions. We derive exact bias bounds, without imposing smoothing assumptions, which are a basis for deriving the rates of convergence under differentiability assumptions.Publicad

A combined measure for quantifying and qualifying the topology preservation of growing self-organizing maps

The Self-OrganizingMap (SOM) is a neural network model that performs an ordered projection of a high dimensional input space in a low-dimensional topological structure. The process in which such mapping is formed is defined by the SOM algorithm, which is a competitive, unsupervised and nonparametric method, since it does not make any assumption about the input data distribution. The feature maps provided by this algorithm have been successfully applied for vector quantization, clustering and high dimensional data visualization processes. However, the initialization of the network topology and the selection of the SOM training parameters are two difficult tasks caused by the unknown distribution of the input signals. A misconfiguration of these parameters can generate a feature map of low-quality, so it is necessary to have some measure of the degree of adaptation of the SOM network to the input data model. The topologypreservation is the most common concept used to implement this measure. Several qualitative and quantitative methods have been proposed for measuring the degree of SOM topologypreservation, particularly using Kohonen's model. In this work, two methods for measuring the topologypreservation of the Growing Cell Structures (GCSs) model are proposed: the topographic function and the topology preserving ma

Global rates of convergence for the bias of singular integral estimators and their shifted versions

This paper provides global rates of convergence for the bias of singular integral estimators of a probability density function under weak conditions, which avoid differentiability. It is also shown that, under smoothing conditions, shifting the singular integral estimators provides a bias reduction technique

Approximation of a stochastic wave equation in dimension three, with application to a support theorem in Hölder norm: the non-stationary case

This paper is a continuation of (Bernoulli 20 (2014) 2169-2216) where we prove a characterization of the support in Hölder norm of the law of the solution to a stochastic wave equation with three-dimensional space variable and null initial conditions. Here, we allow for non-null initial conditions and, therefore, the solution does not possess a stationary property in space. As in (Bernoulli 20 (2014) 2169-2216), the support theorem is a consequence of an approximation result, in the convergence of probability, of a sequence of evolution equations driven by a family of regularizations of the driving noise. However, the method of the proof differs from (Bernoulli 20 (2014) 2169-2216) since arguments based on the stationarity property of the solution cannot be used

On universal unbiasedness of delta estimators

This paper considers delta estimators of a Radon-Nicodym derivative of a probability function with respect to a measure. Sufficient conditions for asymptotic unbiasedness and global rates of convergence, which can be improved by imposing differentiability conditions on the estimated curves, are provided. A bias reduction technique is proposed, and the application of the results to regression estimation is discussed. The sufficient conditions for asymptotic unbiasedness are checked for some broad classes of nonparametric estimators