  We discuss a class of difference-based estimators for the autocovariance in
nonparametric regression when the signal is discontinuous (change-point
regression), possibly highly fluctuating, and the errors form a stationary
$m$-dependent process. These estimators circumvent the explicit pre-estimation
of the unknown regression function, a task which is particularly challenging
for such signals. We provide explicit expressions for their mean squared errors
when the signal function is piecewise constant (segment regression) and the
errors are Gaussian. Based on this we derive biased-optimized estimates which
do not depend on the particular (unknown) autocovariance structure. Notably,
for positively correlated errors, that part of the variance of our estimators
which depends on the signal is minimal as well. Further, we provide sufficient
conditions for $\sqrt{n}$-consistency; this result is extended to piecewise
Holder regression with non-Gaussian errors.
  We combine our biased-optimized autocovariance estimates with a
projection-based approach and derive covariance matrix estimates, a method
which is of independent interest. Several simulation studies as well as an
application to biophysical measurements complement this paper.
