MAST20005/MAST90058: Week 12 Problems
1. Let X1,…,Xn be a random sample from Bi(1,p).
(a) Find the Cram ́er–Rao lower bound for unbiased estimators of p.
(b) We know that X ̄ is an unbiased estimator of p. Show that X ̄ attains the Cram ́er–Rao lower bound.
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2. Let X1,…,Xn be a random sample from N(μ,θ) where μ is known. (a) Show that the maximum likelihood estimator of θ is,
(c) What is the approximate distribution of θ?
(d) What is the exact distribution of nθ/θ?
3. Let X1, . . . , Xn be a random sample from the density:
(Xi − μ)2.
(b) Find the Cram ́er–Rao lower bound for unbiased estimators of θ.
f(x|θ)= xe−x/θ, 0
(b) X∼Unif(−2θ,2θ)
6. Find sufficient statistics for θ (where θ > 0) when we observe X from the following pdfs:
(a) f(x|θ)= 1θe−x/θ, 0