  This paper is concerned with estimating the intersection point of two
densities, given a sample of both of the densities. This problem arises in
classification theory. The main results provide lower bounds for the
probability of the estimation errors to be large on a scale determined by the
inverse cube root of the sample size. As corollaries, we obtain probabilistic
bounds for the prediction error in a classification problem. The key to the
proof is an entropy estimate. The lower bounds are based on bounds for general
estimators, which are applicable in other contexts as well. Furthermore, we
introduce a class of optimal estimators whose errors asymptotically meet the
border permitted by the lower bounds.
