When the amount of information passing through a channel is estimated on the basis of a sample, situations of two kinds can arise: Either (A) the statistical structure of the source of information is unknown or (B) it is known. In the present note only discrete sources and channels without probability aftereffects are discussed. In either kind of situation the estimator proposed is found to be in general asymptotically normal with a variance of the order of the reciprocal of the sample size. There are, however, important exceptions: For some systems (i.e. combinations of channels and matching sources) the estimator in question has a variance of the order of the reciprocal of the squared sample size; it is shown that then the corresponding asymptotic distribution is that of a quadratic form in Gaussian random variables. The nature of these exceptional systems is investigated; they are characterized by some extremal properties. A simplified proof of a classical result of Neyman and Pearson is obtained incidentally. Finally, the situation in which the channel is known is treated briefly at the end.
