  We consider the high order moments estimator of the frontier of a random pair
introduced by Girard, S., Guillou, A., Stupfler, G. (2012). {\it Frontier
estimation with kernel regression on high order moments}. In the present paper,
we show that this estimator is strongly uniformly consistent on compact sets
and its rate of convergence is given when the conditional cumulative
distribution function belongs to the Hall class of distribution functions.
