Sufficient statistic for uniform distribution

Question

Given random sample $\left\{ { X }_{ 1 },{ X }_{ 2 },...,{ X }_{ n } \right\} $ from $ U(0,\theta)$.

Let ${Y}_{i}$ be the order statistics.

Then the sufficient statistic for $\theta$ is ${ Y }_{ n }$.

My question is how to prove it by Factorization Theorem.

I can only factor the joint p.d.f of the random sample to this form,

${ \left( \frac { 1 }{ \theta } \right) }^{ n }{ I }_{ \left[ 0,{ Y }_{ n } \right] }({ y }_{ 1 }){ I }_{ \left[ { Y }_{ 1 },\theta \right] }\left( { y }_{ n } \right) $ where $I$ is the indicator function.

I cannot figure out why this is independent of ${Y}_{1}$.

Need some help. Thanks in advance.

Answer 1

Just figured out!

Factorize it in another way!

${ \left( \frac { 1 }{ \theta } \right) }^{ n }{ I }_{ \left[ { Y }_{ 1 },\theta \right] }\left( { y }_{ n } \right)\prod _{ i=1 }^{ n }{ {I}_{\left[0,\infty\right)} }({x}_{i}) $