On Stability of Linear Estimators in Poisson Noise

This paper considers estimation of a random variable in Poisson noise. Specifically, the main focus is to assess optimality and near optimality conditions for linear estimators.In the first part of the paper, it is shown that linear estimators are optimal if and only if the underlying prior is a gamma distribution and the dark current parameter is zero.In the second part of the paper, a stability analysis of linear estimators is undertaken. Specifically, it is shown that if an optimal estimator is close to a linear estimator in an L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> ,p ≥1 distance, then the underlying prior distribution is approximately gamma in the Lévy metric and the Kolmogorov metric.