  We consider the model of nonregular nonparametric regression where smoothness
constraints are imposed on the regression function $f$ and the regression
errors are assumed to decay with some sharpness level at their endpoints. The
aim of this paper is to construct an adaptive estimator for the regression
function $f$. In contrast to the standard model where local averaging is
fruitful, the nonregular conditions require a substantial different treatment
based on local extreme values. We study this model under the realistic setting
in which both the smoothness degree $\beta>0$ and the sharpness degree
$\mathfrak {a}\in(0,\infty)$ are unknown in advance. We construct adaptation
procedures applying a nested version of Lepski's method and the negative Hill
estimator which show no loss in the convergence rates with respect to the
general $L_q$-risk and a logarithmic loss with respect to the pointwise risk.
Optimality of these rates is proved for $\mathfrak{a}\in(0,\infty)$. Some
numerical simulations and an application to real data are provided.
