The Annals of Statistics
- Ann. Statist.
- Volume 37, Number 6A (2009), 3396-3430.
Wavelet regression in random design with heteroscedastic dependent errors
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, , and for a variety of Lp error measures. We consider error distributions with Long-Range-Dependence parameter α, 0<α≤1; heteroscedasticity is modeled with a design dependent function σ. 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, π and α. Furthermore, we show that long range dependence does not come into play for shape estimation f−∫f. The theory is illustrated with some numerical examples.
Ann. Statist., Volume 37, Number 6A (2009), 3396-3430.
First available in Project Euclid: 17 August 2009
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Kulik, Rafał; Raimondo, Marc. Wavelet regression in random design with heteroscedastic dependent errors. Ann. Statist. 37 (2009), no. 6A, 3396--3430. doi:10.1214/09-AOS684. https://projecteuclid.org/euclid.aos/1250515391