Find sufficient statistic for a mixed distribution

Question

If I have a random variable $Y$ with a mixed distribution: $$ F(y)=\begin{cases} 0, & y<1\\ 1/\theta^3, &y=1\\ y^3/\theta^3, &1<y<\theta\\ 1, &y\le\theta \end{cases} $$ and what I want to do is to find the sufficient statistic for $\theta$. If I do not consider the point where $y=1$, I could get p.d.f. and use Factorization Theorem to get the sufficient statistic as $\max(Y_i)$. But, am I right to ignore that point?

Answer 1

I believe you can use the Factorization Theorem, as the p.d.f can be written : $$\begin{aligned}f(y) &= 2y^2/\theta^3\mathbb{I}_{y>1}\mathbb{I}_{y<\theta} + 1/\theta^3\mathbb{I}_{y=1} \\ &= 2y^2/\theta^3\mathbb{I}_{y>1}\mathbb{I}_{y<\theta} + 1/\theta^3\mathbb{I}_{y=1}\mathbb{I}_{y<\theta} \\ &= \mathbb{I}_{y<\theta}/\theta^3 \left(2y^3\mathbb{I}_{y>1} + \mathbb{I}_{y=1} \right). \end{aligned}$$ Thus, $Y$ is a sufficient statistic.