Hyperbolic Wavelet-Based Methods in Nonparametric Function Estimation and Hypothesis Testing
Jean-Marc Freyermuth, ORSTAT and Leuven Statistics Research Center, K.U.Leuven, Belgium

Seminar • 29 November 2013 • BM 4.233

Abstract
In this talk we are interested in nonparametric multivariate function estimation. In Autin et al. (2012), we determine the maxisets of several estimators based on thresholding of the empirical hyperbolic wavelet coefficients. That is we determine the largest functional space over which the risk of these estimators converges at a chosen rate. It is known from the univariate setting that pooling information from geometric structures (horizontal/vertical blocks) in the coefficient domain allows to get ’large’ maxisets (see e.g Autin et al., 2011a,b,c). In the multidimensional setting, the situation is less straightforward. In a sense these estimators are much more exposed to the curse of dimensionality. However we identify cases where information pooling has a clear benefit. In particular, we identify some general structural constraints that can be related to compound models and to a ’minimal’ level of anisotropy. If time allows we will also discuss either the application of such methods for estimating the time-frequency spectrum of a (zero mean) non-stationary time series with second order structure which varies across time (in the spirit of (Neumann and von Sachs, 1997)); or how the geometry of the hyperbolic wavelet basis allows to construct ’optimal’ testing procedures of some structural characteristics of the estimand.