Mathematics – Statistics Theory
Scientific paper
2009-09-02
Annals of Statistics 2009, Vol. 37, No. 6A, 3396-3430
Mathematics
Statistics Theory
Published in at http://dx.doi.org/10.1214/09-AOS684 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Scientific paper
10.1214/09-AOS684
We investigate function estimation in nonparametric regression models with random design and heteroscedastic correlated noise. Adaptive properties of warped wavelet nonlinear approximations are studied over a wide range of Besov scales, $f\in\mathcal{B}^s_{\pi,r}$, and for a variety of $L^p$ error measures. We consider error distributions with Long-Range-Dependence parameter $\alpha,0<\alpha\leq1$; heteroscedasticity is modeled with a design dependent function $\sigma$. We prescribe a tuning paradigm, under which warped wavelet estimation achieves partial or full adaptivity results with the rates that are shown to be the minimax rates of convergence. For $p>2$, it is seen that there are three rate phases, namely the dense, sparse and long range dependence phase, depending on the relative values of $s,p,\pi$ and $\alpha$. Furthermore, we show that long range dependence does not come into play for shape estimation $f-\int f$. The theory is illustrated with some numerical examples.
Kulik Rafał
Raimondo Marc
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