Suppose we observe independent pairs (Xi,Yi) where each (Xi,Yi) has a uniform distribution in the circle of unknown radius θ and centered at (0,0) in the plane.
Show that for the method of moments estimator and the maximum likelihood estimator, it is the case that the distribution of ˆθ/θ does not depend on θ. Explain why this means we can write
MSEθ(ˆ θ) = θ2*MSE1(ˆ θ) where here the subscript θ means "under the assumption that the true value is θ," so MSE1 denotes the mean square error under the assumption that θ = 1. From this, explain why it suffices that we compare the two estimators when θ = 1