Rutgers University Unbiased Estimators and Efficiency Questions

Homework 6. Unbiased Estimators and EfficiencyDue Thursday, 10/20, by 11 am
(1) X1 , X2 , X3 are iid normal, Xi ∼ N (µ, σ 2 ).
b 1 := X1 +X2 +X3 and Θ
b 2 := X1 +X2 +2X3 are unbiased esti(a) Show that Θ
3
4
mators of µ.
b 1 and Θ
b 2.
(b) Directly compare efficiency of Θ
(2) (a) (X1 , X2 , . . . , Xn ) are iid, Xi has Poisson distribution with parameter
k
λ, i.e. P rob(Xi = k) = e−λ · λk! for k = 0, 1, 2, . . .. Show that X is a
minimal variance unbiased estimator of λ. You can use the facts that
E[Xi ] = λ and V ar[Xi ] = λ.
(b) (X1 , X2 , . . . , Xn ) are iid, Xi has exponential distribution with parameter λ, i.e. has probability density λe−λx for x ≥ 0. Show that X is a
minimal variance unbiased estimator of θ := λ1 . You can use the facts
that E[Xi ] = λ1 and V ar[Xi ] = λ12 .
b
(3) (X1 , X2 , . . . , Xn ) are iid,
h iXi ∼ Bernoulli(p), Θ is a statistic.
b
b
(a) Show that lim E Θ = Θ(0, 0, . . . , 0).
| {z }
p→0
n
b can’t be an unbiased estimator of θ := 1−p .
(b) Show that Θ
p
(4) (X1 , X2 , . . . , Xn ) are iid uniform, Xi ∼ U [0, θ]. Yk is the k-th largest value
of X1 , X2 , . . . , Xn . You can use the fact that Yk /θ has probability density
n!
k−1
(1 − x)n−k for 0 ≤ x ≤ 1.
(k−1)!·(n−k)! x
b k := n+1 · Yk is an unbiased estimator of θ. You will need
(a) Show that Θ
k
to derive E[Yk ].
b 1, Θ
b 2, . . . , Θ
b n by efficiency. You will need to derive V ar[Yk ].
(b) Compare Θ
(5) S is a random sample of size n taken without replacement from a finite
population of positive integers {1, 2, . . . , N }. Let Y := max S.
(m−1)
(a) Show that P rob(Y = m) = n−1
.
(Nn )
(b) Show that n+1
n · Y − 1 is an unbiased estimator of N . You will need
to derive E[Y ]
1