Method of Moments

Introduction to the Science of Statistics
The Method of Moments
A (↵, ) random variable has mean ↵/ and variance ↵/ 2 . Because we have two parameters, the method of
moments methodology requires us to determine the first two moments.
E (↵, ) X 1 = ↵ and E (↵, ) X 2 1 = Var (↵, ) (X 1 )+E (↵, ) [X 1 ] 2 = ↵ 2 + ✓ ↵
◆ 2
=
Thus, for step 1, we find that
↵(1 + ↵)
2
= ↵ 2 + ↵2 2 .
µ 1 = k 1 (↵, )= ↵ , µ 2 = k 2 (↵, )= ↵ 2 + ↵2 2 .
For step 2, we solve for ↵ and
. Note that
µ 2 µ 2 1 = ↵ 2 ,
and
So set
to obtain estimators
µ 1
µ 1 ·
µ 2 µ 2 1
ˆ =
¯X = 1 n
µ 1
µ 2 µ 2 1
= ↵/
↵/ 2 = ,
= ↵ · = ↵, or ↵ = µ2 1
µ 2 µ 2 .
1
nX
X i and X 2 = 1 n
i=1
nX
i=1
X 2 i
¯X
X 2 ( ¯X)
and ˆ↵ = ˆ ¯X (
=
¯X) 2
2 X 2 ( ¯X) . 2
dgamma(x, 0.23, 5.35)
0 2 4 6 8 10 12
0.0 0.2 0.4 0.6 0.8 1.0
x
Figure 13.1: The density of a
(0.23, 5.35) random variable.
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