# Proving sufficiency of a statistic using the expectation

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?

• Sorry, I made a mistake when I transcribed the question. I have just modified it.
– QGM
Feb 16, 2021 at 18:50
• Thanks again. In fact you are right, there is a dependance of $f_W$ in $\theta$ that I had not made explicit.
– QGM
Feb 16, 2021 at 19:17
• This last point is what is not clear for me
– QGM
Feb 16, 2021 at 19:24

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.