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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.Īn elaborate theory of predictions of a causal hypothesis consists of several falsifiable statements derived from the causal hypothesis. Second, the optimally adaptive local rates are in general not the same in fixed lattice and random designs. First, the max–min block estimator automatically adapts to the local smoothness and the intrinsic dimension of the isotonic regression function at the optimal rate. There are two interesting features in our local theory. samples in $d$ dimensions can have a nonstandard $\mathcal$ in a local asymptotic minimax sense. Such mis-specified settings can lead to singularity in the Fisher information matrix, and moreover, the maximum likelihood estimator based on $n$ i.i.d. We consider over-specified settings in which the number of fitted components is larger than the number of components in the true distribution. Examples include suitably separated Gaussian mixture models and mixtures of linear regressions. A line of recent work has analyzed the behavior of the Expectation-Maximization (EM) algorithm in the well-specified setting, in which the population likelihood is locally strongly concave around its maximizing argument.