In: Statistics and Probability
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.