Suppose the model $P_θ$ is the class of all continuous distributions; this is called a ‘nonparametric family’, where the unknown parameter $θ$ is the whole distribution function. Let $x_1,...,x_n$ be an iid sample from $P_θ$. Show that the order statistics are sufficient for $P_θ$.
My thinking:
given $\theta$ is the whole distribution function of order statistics we have:
$$F_{\Theta_{(r)}}(\theta) = \sum_{j=r}^n\binom{n}{j}F_{\Theta}(\theta)^j(1-F_{\Theta}(\theta))^{n-j}$$
and by the factorisation theorem we have: $$\left\{F_{\Theta}(\theta)^j(1-F_{\Theta}(\theta))^{n-j}\right\}\left\{\sum_{j=r}^n\binom{n}{j}\right\}$$
such that $p_{\theta}(x) = g(t(x), \theta)h(x)$ where $h(x)$ is free of $\theta$. In this case, we have that for the order statistics $\left\{F_{\Theta}(\theta)^j(1-F_{\Theta}(\theta))^{n-j}\right\}$ is sufficient?
On another note: what is meant by $P_{\theta}$ is the "class"?