Apparent inconsistency arising from showing that $x_{(n)}$ is sufficient for $\theta$ where $X \sim \frac{1}{\theta}\mathbb{I}_{(0, \theta)}$

Question

The problem is to show that the largest order statistic $x_{(n)}$ is sufficient for $\theta$ where $X \sim \frac{1}{\theta}\mathbb{I}_{x \in (0, \theta)}$ is a uniform distribution.

I believe I have proved the result, but my argument cannot make sense, due to the consequences. Here's my argument:

The likelihood of a random sample of size $n$ is, for $\theta >0$, $$L(\theta) = \bigg(\frac{1}{\theta} \bigg)\bigg(\frac{1}{\theta} \bigg) \cdots \bigg(\frac{1}{\theta} \bigg) = \frac{1}{\theta^n},$$ which factors into a product of a function of $\theta$ and $\hat\theta$, and a function of the sample $x_i$ for any statistic $\hat\theta$. Therefore, by the Fisher-Neyman Factorization Theorem, any statistic is sufficient for $\theta$. In particular, the largest order statistic $x_{(n)}$ is sufficient for $\theta$.

It seems to me like any statistic should be sufficient for $\theta$. Of course, "any" statistic includes a constant, so my argument must be wrong. I'm not sure why, though. Can anyone enlighten me?

Thanks

Answer 1

The problem is in the way you wrote the likelihood function. Suppose that we observe $x_{(n)}=2$, where $x_{(n)}$ is the $n$th ordered value of the sample $x_1,\dots,x_n$. Then what is $L(1)$? According to your formula it would be 1, but $\theta$ being 1 is not likely at all since we observe a draw from $U(0,\theta)$ that is higher than 1. Therefore, $L(1)=0$.

That is, for $\theta$ to be feasible, i.e. to have a positive likelihood, we cannot observe $x_{(n)}>\theta$ drawn from $U(0,\theta)$. In other words, by recalling that the density function of $U(0,\theta)$ actually is $f(x)=\frac1\theta 1_{(0,\theta)}(x)$, we get

$$L(\theta)=\frac1\theta1_{(0,\theta)}(x_1)\dots \frac1\theta1_{(0,\theta)}(x_n)=\frac{1}{\theta^n}1_{(0,\theta)}(\max\{x_1,\dots,x_2\})=\frac{1}{\theta^n}1_{(0,\theta)}(x_{(n)}),$$

in which case not any statistic is sufficient.