  We study the problem of estimating piecewise monotone vectors. This problem
can be seen as a generalization of the isotonic regression that allows a small
number of order-violating changepoints. We focus mainly on the performance of
the nearly-isotonic regression proposed by Tibshirani et al. (2011). We derive
risk bounds for the nearly-isotonic regression estimators that are adaptive to
piecewise monotone signals. The estimator achieves a near minimax convergence
rate over certain classes of piecewise monotone signals under a weak
assumption. Furthermore, we present an algorithm that can be applied to the
nearly-isotonic type estimators on general weighted graphs. The simulation
results suggest that the nearly-isotonic regression performs as well as the
ideal estimator that knows the true positions of changepoints.
