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This paper is concerned with estimating the intersection point of twodensities, given a sample of both of the densities. This problem arises inclassification theory. The main results provide lower bounds for theprobability of the estimation errors to be large on a scale determined by theinverse cube root of the sample size. As corollaries, we obtain probabilisticbounds for the prediction error in a classification problem. The key to theproof is an entropy estimate. The lower bounds are based on bounds for generalestimators, which are applicable in other contexts as well. Furthermore, weintroduce a class of optimal estimators whose errors asymptotically meet theborder permitted by the lower bounds.

Optimal third root asymptotic bounds in the statistical estimation of thresholds