On estimation for a binary-symmetric channel

We consider a binary-symmetric channel where the input, modeled as an infinite sequence of bits, is distorted by Bernoulli noise. In a paper by Simons (1991), a consistent estimator of the distortion, i.e., of the probability that a single bit is changed, is described under the basic assumption that the complexity of the input is finite. Here, we deal with two shortcomings. (1) The Simons' estimator requires some information on the value of the complexity; we describe a modified approach through the generalized Bernoulli components in which determination of the complexity is not needed. (2) The Simons' estimator exhibits a serious negative bias when the distortion is great; our modification largely overcomes this deficiency while reducing the variance. The asymptotics for the proposed estimators are studied, and their properties are illustrated through numerical studies.