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• W3029205332 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 nontrivial answer to an open problem in the shape-constrained analysis literature. The minimax rate under the loss of the sum of squared errors 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 $theta ^{*}=(theta^{*}_{1},dots ,theta ^{*}_{n})^{mathsf{T}}$. 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." @default.
• W3029205332 created "2020-06-05" @default.
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• W3029205332 date "2020-04-01" @default.
• W3029205332 modified "2023-10-18" @default.
• W3029205332 title "On estimation of isotonic piecewise constant signals" @default.
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• W3029205332 doi "https://doi.org/10.1214/18-aos1792" @default.
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