  We focus on the problem estimating a monotone trend function under additive
and dependent noise. New point-wise confidence interval estimators under both
short- and long-range dependent errors are introduced and studied. These
intervals are obtained via the method of inversion of certain discrepancy
statistics arising in hypothesis testing problems. The advantage of this
approach is that it avoids the estimation of nuisance parameters such as the
derivative of the unknown function, which existing methods are forced to deal
with. While the methodology is motivated by earlier work in the independent
context, the dependence of the errors, especially longrange dependence leads to
new challenges, such as the study of convex minorants of drifted fractional
Brownian motion that may be of independent interest. We also unravel a new
family of universal limit distributions (and tabulate selected quantiles) that
can henceforth be used for inference in monotone function problems involving
dependence.
