Minimal Sufficient Statistic(MSS)

Def)
A sufficient statistic $T(\underline{X})$ is called minimal sufficient statistic if $T(\underline{X})$ is a function of any other sufficient statistic.

Theorem)
If $\frac{\prod f(x;\theta)}{\prod f(y;\theta)}$ NOT depend on $\theta$ <=> $T(\underline{X})=T(\underline{Y})$,
$T(\underline{X})$ is MSS
pf)

ex1)

Ancillary Statistic(AS)

Def)
If the distribution of a statistic $S(\underline{X})$ does not depend on $\theta$,
$S(\underline{X})$ is called ancillary statistic

ex1)

ex2)

Complete Statistic(CS)

Def)
Let $f(t;\theta)$ be a family of pdfs or pmfs for statistic $T(\underline{X})$.
$T(\underline{X})$ is called a complete statistic if $E_{\theta}(g(T))=0$ for all $\theta$ implies $P_{\theta}(g(T)=0)=1$ for all $\theta$
($E_{\theta}(g(T))=0$ 이면 $g(T)=0$ 임을 보여주면 됨.)

ex1)

ex2)

Useful theorems

Theorem)
If $f(x;\underline{\theta})$ belongs to exponential family s.t. $f(x;\underline{\theta})=h(x) c(\underline{\theta}) exp(\sum_{j=1}^{k} w_j(\underline{\theta}) t_j(x))$,
$(\sum_i t_1(x_i),...,\sum_i t_k(x_i))$: complete sufficient statistic
pf)

Theorem) Basu's theorem
If $Y$ is a complete sufficient statistic for $\theta$ and $Z$ is a ancillary statistic for $\theta$,
$Y$ and $Z$ are independent.
pf)