Maximum Likelihood Estimation for Binomially Distributed Signals in Discrete Noise.

Let X be a discrete variable distributed like the sum of independent variables Y and Z, where the signal Y is binomially distributed and the noise variable Z is a nonnegative integer valued variable whose distribution does not depend on the binomial parameter p. The family of convoluted binomial distributions, that is, distributions of variables such as X, are characterized in terms of a system of differential equations satisfied by their probability mass functions. A monotonicity property for probability ratios of certain convoluted binomial distributions is noted, and the maximum likelihood estimate of the binomial parameter is shown to be easily obtained for these distributions using the characterizing property established for their mass functions. Results are applicable to models for binomial signals in noise in which the noise distribution is known or can be estimated from an auxiliary experiment. Author