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), ∑n i=1 Xi ) is jointly sufficient for (?1, ?2).

b) Determine the pdf of X(1).

c) Determine E[X(1)].

d) Determine E[X^2 (1) ].

e) Determine Var[X(1)].

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

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

X(1) is the minimum value. I mostly need help in starting this problem.

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orchestra answered 1 year ago

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