Proving sufficiency of a statistic using the expectation

Question

I am blocked trying to solve the following question. I would appreciate if someone could give me a hint.

Let $X_1,\ldots,X_n$ be $n$ independent random variables following a continuous uniform distribution $U(0,\theta)$, with $\theta>0$. Let $W=\max(X_1,\ldots,X_n)$. I have proved that, for any measurable and bounded functions $h:\mathbb{R}^n\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$, $$\mathbb{E}_\theta[h(X_1,\ldots,X_n)g(W)]=\int_{-\infty}^{\infty}H(s)g(s)f_W(s;\theta)ds,\tag{1}$$ where $f_W(s)$ is the density function of the random variable $W$ and $H(s)$ is a function which does not depend on $\theta$. I would like to deduce from (1) the sufficiency of $W$ (which is intuitively clear for me) but I get stuck in all my trials. I have tried by writing the the distribution functions in terms of the expectation of an indicator function conditioned to the value of $W$ and then applying the properties of the conditioned expectation, but I cannot get something clear.

Could anyone give me some ideas?

Answer 1

The equation (1) shows that, for any measurable function $h:\mathbb R^n\mapsto\mathbb R$, $$\mathbb E_\theta^W[\mathbb E_\theta\{h(X_1,\ldots,X_n)|W\}]=\mathbb E_\theta^W[ H(W)]$$ hence that, almost surely, for any measurable function $h:\mathbb R^n\mapsto\mathbb R$, $$\mathbb E_\theta\{h(X_1,\ldots,X_n)|W\} = H(W)$$ is constant in $\theta$. This implies that the conditional distribution of $(X_1,\ldots,X_N)$ given $W$ is constant in $\theta$, which is the original definition of sufficiency.