$T$ is sufficient for $\mathcal{P}$, sufficient for $\theta$ , sufficient for $F$

Question

The definition which is given in my book about sufficiency is:

A Statistic $T$ is said to be sufficient for the statistical model $ \mathcal{P}= \{P_{\theta} : \theta \in \Theta \}$ of $\boldsymbol{X}$ if the conditional distribution of $\boldsymbol{X}$ given $T= t$ is independent of $\theta$ for all $t$

Now, by the factorization criterion; which in my book states that;

A Statistic $T$ i sufficient for $\mathcal{P}$ iff there exists nonnegative functions $g(\cdot;\theta)$ and $h$ such that the probability functions $p(\cdot ; \theta)$ satisfy: $p(\boldsymbol{x};\theta ) = g(T(\boldsymbol{x});\theta)h(\boldsymbol{x})$

Ok fine, but one corollary states

A statistic $T$ is sufficient for $\theta \in \Theta$ iff $L(\theta;\boldsymbol{x}) \propto g(T(\boldsymbol{x});\theta)$

The proof for this is by utilizing the factorization theorem:

$$\begin{equation} L(\theta;\boldsymbol{x}) = p(\boldsymbol{x};\theta)= g(T(\boldsymbol{x});\theta)h(\boldsymbol{x}) \propto g(T(\boldsymbol{x});\theta) \end{equation}$$

And Another corollary states

Suppose $\boldsymbol{X} = (X_{1},X_{2},..X_{n})$ is sample of i.i.d r.v:s with distribution $F$. Then the order statistic $(X_{[1]},X_{[2]},..,X_{[n]} ) $ is sufficient for $F$

Can someone explain what is the difference between Sufficient Statistic for $\theta$,Sufficient statistic for $F$ and sufficient statistic for $\mathcal{P}$

(maybe then i can understand the proof for the first corollary above, now Iam just a bit confused.)

Answer 1

I think it's mostly a stylistic difference. Generally you talk about sufficiency with regard to a parameter $\theta$, but if $\theta$ somehow indexes the full family of distributions $\mathcal{P} = \{ P_\theta : \theta \in \Theta \}$, then you could just as easily say a sufficient statistic for $\theta$ is sufficient for the entire model. I wouldn't pay too much attention to any particular authors preferences and just remember the important parts from the formal definition about conditioning, and understand intuitively what sufficiency means.

The last corollary is of course true but not very interesting. It's saying that an ordered sample contains all the information from an i.i.d. sample about the distribution $F$ from which it came, which is pretty much a tautology.