An ecologist takes data (xi, yi), i = 1,...,n, where xi is the size of an area and Yi is the number of moss plants in the area. We model the data by Yi ~ Poisson(θxi), Yis independent.
(a) Show that the least squares estimator of θ is ∑xiYi / ∑x2i. Show that this estimator has variance θ∑x3i / (∑x2i)2. Also, compute its bias.
(b) Show that the MLE of θ is ∑Yi / ∑xi has variance θ / ∑xi. Compute its bias.
(c) Find a best unbiased estimator of θ and show that its variance attains the Cramer-Rao Lower Bound.