Empirical Bayesian test of the smoothness.

In the context of adaptive nonparametric curve estimation problem, a common assumption is that a function (signal) to estimate belongs to a nested family of functional classes, parameterized by a quantity which often has a meaning of smoothness amount. It has already been realized by many that the problem of estimating the smoothness is not sensible. What then can be inferred about the smoothness? The paper attempts to answer this question. We consider the implications of our results to hypothesis testing. We also relate them to the problem of adaptive estimation. The test statistic is based on the marginalized maximum likelihood estimator of the smoothness for an appropriate prior distribution on the unknown signal

Local inference by penalization method for biclustering model

We study the problem of inference (estimation and uncertainty quantification problems) on the unknown parameter in the biclustering model by using the penalization method. The underlying biclustering structure is that the high-dimensional parameter consists of a few blocks of equal coordinates. The quality of the inference procedures is measured by the local quantity, the oracle rate, which is the best trade-off between the approximation error by a biclustering structure and the complexity of that approximating biclustering structure. The approach is also robust in that the additive errors are assumed to satisfy only certain mild condition (allowing non-iid errors with unknown joint distribution). By using the penalization method, we construct a confidence set and establish its local (oracle) optimality. Interestingly, as we demonstrate, there is (almost) no deceptiveness issue for the uncertainty quantification problem in the biclustering model. Adaptive minimax results for the biclustering, stochastic block model (with implications for network modeling) and graphon scales follow from our local results

METHODS OF SIGNAL PROCESSING Adaptive Filtering of a Random Signal in Gaussian White Noise

Abstract—We consider the problem of estimating an infinite-dimensional vector θ observed in Gaussian white noise. Under the condition that components of the vector have a Gaussian prior distribution that depends on an unknown parameter β, we construct an adaptive estimator with respect to β. The proposed method of estimation is based on the empirical Bayes approach. DOI: 10.1134/S0032946008040054 1