Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )

Stat 312: Lecture 04
Moment Matching
Moo K. Chung
[email protected]
September 14, 2004
1. Given a random sample X
1
, · · · , X
n
, a linear
estimator of parameter θ is an estimator of
form
ˆ
θ =
n X
i =1
c
i
X
i
.
Then it can be shown that
¯
X is the MVUE
for population mean among all linear unbi-
ased estimators.
Proof. Case n = 2 will be proved. The gen-
eral statement follows inductively. Consider
linear estimators
ˆ μ = c
1
X
1
+ c
2
X
2
.
To be unbiased, c
1
+ c
2
= 1 . To be most ef-
ficient among all unbiased linear estimators,
the variance has to be minimized. The vari-
ance is
V ˆ μ = c
2
1
V X
1
+ c
2
2
V X
2
=
£
c
2
1
+ (1 - c
1
)
2
/
σ
2
The quadratic term in the bracket 2 c 2
1
- 2 c
1
+
1 is minimized when c
1
= 1 / 2 .
2. Given a random sample X
1
, · · · , X
n
, the k -
th sample moment is M
k
=
∑
n
j =1
X
k
j
/n .
The moment estimators of population param-
eters are obtained by matching the sample
moments to correspond population moments
and solving the resulting equations simulta-