3: Uniform distribution Let X!, , Xn be iid. from a uniform(0,0) distribution where θ is unknown, that is 0, otherwise a) Find the maximum likelihood estimator for 0, 0MLE b) Explain why will be a sufficient and complete statistic for 0 (see example 6.2.23) c) Find the method of moments estimator for θ.0MME- Is it a function of T(X)? d) Find the expectation and variance of 0MME e) Start with θΜΜΕ, and use the Rao-Blackwell theorem to construct a new estimator for θ, θRBE-What do you know about this estimator? f) Show that the probability density function of T(X) is given by 0 otherwise Use this distribution to find the expectation and variance for θMLE and θ11E g) Compare the MSE for MME, 0MLE and ORB h) Simulate n = 7 independent uniformly distributed values on (0,3) and cornpute 0MME and θRBE Repeat this experiment B 1000 times, and compare the performance of the two estima- tors by means, variances, histograms etc. (Hint: In R you can use function runif(7,0,3) to simulate an experiment.)

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