PDF of Lecture Notes - School of Mathematical Sciences

2. STATISTICAL INFERENCE
Example
X 1 , . . . , X n ∼ i.i.d. Exp(λ) =⇒ E(X i ) = 1 ; the method of moments estimator is
λ
defined as the solution to the equation
¯X = 1˜λ =⇒ ˜λ = 1¯X .
Remark
The method of moments is appealing:
(1) it is simple;
(2) rationale is that ¯X is BLUE for µ.
If θ = (θ 1 , θ 2 , . . . , θ p ), the method of moments can be adapted as follows:
(1) let µ k = µ k (θ) = E(X k ), k = 1, . . . , p;
(2) let m k = 1 n
n∑
x k i .
i=1
The MoM estimator ˜θ is defined to be the solution to the system of equations
m 1 = µ 1 (˜θ)
m 2 = µ 2 (˜θ)
.
.
m p = µ p (˜θ)
Example
Suppose X 1 , X 2 , . . . , X n are i.i.d. N(µ, σ 2 ) & let θ = (µ, σ 2 ).
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