  We introduce a new estimator SMUCE (simultaneous multiscale change-point
estimator) for the change-point problem in exponential family regression. An
unknown step function is estimated by minimizing the number of change-points
over the acceptance region of a multiscale test at a level \alpha. The
probability of overestimating the true number of change-points K is controlled
by the asymptotic null distribution of the multiscale test statistic. Further,
we derive exponential bounds for the probability of underestimating K. By
balancing these quantities, \alpha will be chosen such that the probability of
correctly estimating K is maximized. All results are even non-asymptotic for
the normal case. Based on the aforementioned bounds, we construct
asymptotically honest confidence sets for the unknown step function and its
change-points. At the same time, we obtain exponential bounds for estimating
the change-point locations which for example yield the minimax rate O(1/n) up
to a log term. Finally, SMUCE asymptotically achieves the optimal detection
rate of vanishing signals. We illustrate how dynamic programming techniques can
be employed for efficient computation of estimators and confidence regions. The
performance of the proposed multiscale approach is illustrated by simulations
and in two cutting-edge applications from genetic engineering and photoemission
spectroscopy.
