Exposé (.pdf) : : 9th International Conference of the ERCIM Working Group on Computational and Methodological Statistics à Seville en Espagne

Présenté par :

Jan JOHANNES

Titre :

Adaptive aggregation in circular deconvolution in the presence of dependence

Abstrait :

A circular deconvolution problem is considered, where the density of a circular random variable X has to be estimated nonparametrically from a noisy observation Y of X. The additive measurement error is supposed to be independent of X and its density is known. The objective is the construction of a fully data-driven estimation procedure based on a dependent sample from Y. Assuming an independent and identically distributed (iid) sample it has been shown in Johannes et al. (2015) that an orthogonal series estimator with adaptive choice of the dimension parameter using a model selection approach can attain minimax-optimal rates of convergence up to a constant. We propose a fully data-driven shrinkage estimator which is inspired by a Bayes estimator in an indirect sequence space model with hierarchical prior and can be interpreted as a fully data-driven aggregation of the orthogonal series estimators. Considering first an iid sample we show that the fully data-driven shrinkage estimator can attain minimax-optimal rates of convergence over a wide range of density classes covering in particular ordinary and super smooth densities. Dismissing then the independence condition and assuming sufficiently weak dependence characterised by a fast decay of the mixing coefficients we derive an upper risk bound for the shrinkage estimator which coincides up to constant with the minimax-optimal risk bound in the iid case.
Références :

Johannes, J., Simoni, A. and Schenk, R. (2015). Adaptive Bayesian estimation in indirect Gaussian sequence space models. Discussion paper, arXiv:1502.00184.

Comte, F., Johannes, J., and Loizeau, X. (2016). Adaptive aggregation in circular deconvolution in the presence of dependence. Discussion paper in preparation.