  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.
