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Let X1,…, Xn be a sample of iid Exp(?1, ?2) random variables with common pdf f...

Let X1,…, Xn be a sample of iid Exp(?1, ?2) random variables with common pdf f (x; ?1, ?2) = (1/?1)e−(x−?2)/?1 for x > ?2 and Θ = ℝ × ℝ+.

a) Show that S = (X(1), ∑ni=1 Xi ) is jointly sufficient for (?1, ?2).

b) Determine the pdf of X(1).

c) Determine E[X(1)].

d) Determine E[X2(1) ].

e ) Is X(1) an MSE-consistent estimator of ?2?

f) Given S = (X(1), ∑ni=1 Xi )is a complete sufficient statistic for(?1, ?2), determine the UMVUEs of ?1 and ?2.

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