PDF of Lecture Notes - School of Mathematical Sciences

2. STATISTICAL INFERENCE
Example
Suppose x 1 , x 2 , . . . , x n are i.i.d. Bernoulli-θ and let s =
θ.
n∑
x i . Then S is sufficient for
i=1
Proof.
P (x) =
n∏
θ x i
(1 − θ) 1−x i
i=1
= θ P x i
(1 − θ) P (1−x i )
= θ s (1 − θ) n−s .
Next observe that S ∼ B(n, θ)
=⇒ P (s) =
=⇒ P (x|s) =
=
=
( n
s)
θ s (1 − θ) n−s
P ({X = x} ∩ {S = s})
P (S = s)
P (X = x)
P (S = s)
⎧
1
( if ⎪⎨ n
s)
⎪⎩
n∑
x i = s
i=1
0, otherwise.
Theorem. 2.2.4 The Factorization Theorem
Suppose x 1 , . . . , x n have joint PDF/prob function f(x; θ). Then S is a sufficient statistic
for θ if and only if
f(x; θ) = g(s; θ)h(x)
for some functions g, h.
Proof.
Omitted.
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