Title: On Estimation of Isotonic Piecewise Constant Signals,
Abstract: Consider a sequence of real data points $X_1,\ldots, X_n$ with underlying
means $\theta^*_1,\dots,\theta^*_n$. This paper starts from studying the
setting that $\theta^*_i$ is both piecewise constant and monotone as a function
of the index $i$. For this, we establish the exact minimax rate of estimating
such monotone functions, and thus give a non-trivial answer to an open problem
in the shape-constrained analysis literature. The minimax rate involves an
interesting iterated logarithmic dependence on the dimension, a phenomenon that
is revealed through characterizing the interplay between the isotonic shape
constraint and model selection complexity. We then develop a penalized
least-squares procedure for estimating the vector
$\theta^*=(\theta^*_1,\dots,\theta^*_n)^T$. This estimator is shown to achieve
the derived minimax rate adaptively. For the proposed estimator, we further
allow the model to be misspecified and derive oracle inequalities with the
optimal rates, and show there exists a computationally efficient algorithm to
compute the exact solution.