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Abstract:
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We study limit distributions for the tuning-free max-min block estimator originally proposed in \cite{fokianos2017integrated} in the problem of multiple isotonic regression, under both fixed lattice design and random design settings. The limiting distribution depends on the the local smoothness of the regression function and the design points, and generalizes the well-known Chernoff distribution in univariate problems.
There are two interesting features in our local theory. First, the max-min block estimator automatically adapts to the full spectrum of local smoothness levels and the intrinsic dimension of the isotonic regression function at the optimal rate. Second, the optimally adaptive local rates are in general not the same in fixed lattice and random designs. In fact, the local rate in the fixed lattice design case is no slower than that in the random design case, and can be much faster when the local smoothness levels of the isotonic regression function or the sizes of the lattice differ substantially along different dimensions.
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