Suppose we choose a random sample of size n from a population with pdf f X (x)=3 ? 3 x 4 for x>? . Note that E( X )=1.5? and Var ( X )=0.75? 2 .

a) (2) Note that the MME for ? is T1=2 X /3 . Is this estimator unbiased for ? ?

b) (3) Consider an estimator T2=( 3n?1 3n ) min X i , where min Xi is the smallest observation. Show that this estimator is unbiased for ? . Hint: first find E(min X i ) using f Y =min Xi (y)=3n ? 3n y ?3n?1 for y>? .

c) (2) Determine Var (T1 ) .

d) (2) Given that E(T2 2 )=? 2 (3n?1) 2 3n(3n?2) , determine Var (T2 ) .

e) (2) Determine the relative efficiency of T1 with respect to T2 . Which one is the more efficient estimator?

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