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Entropy estimate for high-dimensional monotonic functions. (English) Zbl 1221.62008

Summary: We establish upper and lower bounds for the metric entropy and bracketing entropy of the class of \(d\)-dimensional bounded monotonic functions under \(L^p\) norms. It is interesting to see that both the metric entropy and bracketing entropy have different behaviors for \(p<d/(d-1)\) and \(p>d/(d-1)\). We apply the new bounds for the bracketing entropy to establish a global rate of convergence of the MLE of a \(d\)-dimensional monotone density.

MSC:

62B10 Statistical aspects of information-theoretic topics
62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference
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