Consider a sequence of data points @%X_1,...,X_n%@ whose underlying mean is @%\theta^*\in\mathbb{R}^n%@. This paper
studies the setting where @%\theta^*%@ is both piecewise constant and monotone. Our contributions are two-fold. First, 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. Secondly, a penalized least-squares procedure is developed for estimating @%\theta^*%@ without knowing the underlying number of pieces. The estimator is shown to achieve the derived minimax rate adaptively. A computationally efficient algorithm is also proposed to compute the exact solution of the penalized least-
squares estimator. The techniques developed in the proofs are of independent interest and applicable to the study of other shape-constrained estimation problems. This talk is based
