## 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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### References:

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