PDF of Lecture Notes - School of Mathematical Sciences

2. STATISTICAL INFERENCE
=⇒ f is an exponential family with
A(λ) = −λ
B(λ) = log λ
t(x) = x
h(x) = 0
We can also check that E(X) = −B′ (λ)
A ′ (λ) .
E(X) = 1 λ
for X ∼ Exp(λ).
In particular, we have seen previously
Now observe that A ′ (λ) = −1, B ′ (λ) = 1 λ
=⇒ −B′ (λ)
A ′ (λ)
= 1 , as required.
λ
It also follows that if x 1 , x 2 , . . . , x n are i.i.d. Exp(λ) observations, then ¯X = 1 n
is the MVUE for 1 λ = E(X).
Definition. 2.2.4
n∑
t(x i )
i=1
Consider data with PDF/prob. function, f(x; θ). A statistic, S(x), is called a sufficient
statistic for θ if f(x|s; θ) does not depend on θ for all s.
Remarks
(1) We will see that sufficient statistics capture all of the information in the
data x that is relevant to θ.
(2) If we consider vector-valued statistics, then this definition admits trivial
examples, such as s = x
⎧
⎨ 1 x = s
since P (X = x|S = s) =
⎩
0 otherwise
which does not depend on θ.
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