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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.

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.)

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 $\theta$, $F$ and $\mathcal{P}$ maybe then i can understand the proof for the first corollary above, now Iam just a bit confused.

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.

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:

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

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 $\theta$, $F$ and $\mathcal{P}$ maybe then i can understand the proof for the first corollary above, now Iam just a bit confused.

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 $\theta$, $F$ and $\mathcal{P}$ maybe then i can understand the proof for the first corollary above, now Iam just a bit confused.