(Solution) - An ecologist takes data xi yi i 1 n where -(2025 Original AI-Free Solution)

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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(?xi), Yis independent.
(a) Show that the least squares estimator of ? is ?xiYi / ?x2i. Show that this estimator has variance ??x3i / (?x2i)2. Also, compute its bias.
(b) Show that the MLE of ? is ?Yi / ?xi has variance ? / ?xi. Compute its bias.
(c) Find a best unbiased estimator of ? and show that its variance attains the Cramer-Rao Lower Bound.