2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/130941We 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.11 pages. Submitted to Journal of Multivariate Analysis. See also http://www.stat.washington.edu/jaw/RESEARCH/PAPERS/available.htmlStatistics TheoryClassical Analysis and ODEs41A46, 46E35 (primary), 62G05, 62G07 (secondary)text