PDF of Lecture Notes - School of Mathematical Sciences

2. STATISTICAL INFERENCE
(3) If t(x) is a sufficient statistic for θ, then ˆθ depends on the data only as a
function of t(x).
Proof. By the Factorization Theorem, t(x) is sufficient for θ iff
f(x; θ) = g(t(x); θ)h(x)
=⇒ l(θ; x) = log g(t(x); θ) + log h(x)
=⇒ ˆθ maximizes l(θ; x) ⇔ ˆθ maximizes log g(t(x); θ)
=⇒ ˆθ is a function of t(x).
Example
Suppose X 1 , X 2 , . . . , X n are i.i.d. Exp(λ); then
f X (x; λ) =
n∏
i=1
λe −λx i
= λ n e −λ P n
i=1 x i
= λ n e −λn¯x .
By the Factorization Theorem, ¯x is sufficient for λ. To get the MLE,
U(λ, x) = ∂l
∂λ = ∂ (n log λ − nλ¯x)
∂λ
= n λ − n¯x;
∂l
∂λ = 0 =⇒ 1 λ = ¯x =⇒ ˆλ = 1¯x .
Note: as proved ˆλ is a function of the sufficient statistic ¯x.
Let Y 1 , Y 2 , . . . , Y n be defined by Y i = log X i .
99