Entropy estimate for high-dimensional monotonic functions

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 bracketing entropy to establish a global rate of convergence of the MLE of a d-dimensional monotone density.