1. No category

6.1-13 Let X1 , X2 , · · · , Xn be a random sample from a distribution having finite variance σ 2 .
Show that
n
X
(Xi − X)2
2
S =
n−1
i=1
is an unbiased estimator of σ 2 .
Solution: We need to show that E(S 2 ) = σ 2 . Recall that S 2 can be expressed as
n
1 X 2
2
S =
X − nX .
n − 1 i=1 i
2
We obtain
n
h X
i
1
2
E
Xi2 − nX
n−1
i=1
n
h
i
X
1
2
E(Xi2 ) − nE(X )
=
n − 1 i=1
n
i
1 hX 2
2
2
=
(σ + µ ) − nE(X ) .
n − 1 i=1
E(S 2 ) =
2
Since E(X ) = E(
Pn
i=1
n
Xi 2
) =
n
P
2
E( n
i=1 Xi )
n2
and (
Pn
i=1
Xi )2 =
Pn
2
i=1 (Xi )
+
P
i6=j
Xi Xj ,
n
i
i
X
X
1 hX
1 hX
E(X ) = 2 E
(Xi )2 +
Xi X j = 2
E(Xi )2 +
EXi Xj .
n
n i=1
i=1
i6=j
i6=j
2
When i 6= j, Xi and Xj are independent and thus EXi Xj = EXi EXj = µ2 . So
n
i 1
X i
1 hX 2
1h
2
2
2
2
2
µ = 2 n(σ + µ ) + n(n − 1)µ = σ 2 + µ2 .
E(X ) = 2
(σ + µ ) +
n i=1
n
n
i6=j
2
Therefore,
" n
#
X
1
1
E(S 2 ) =
(σ 2 + µ2 ) − n( σ 2 + µ2 )
n − 1 i=1
n
1 =
n(σ 2 + µ2 ) − (σ 2 + nµ2 )
n−1
1 =
(n − 1)σ 2
n−1
= σ2.