A nonnegative random variable $x$ has a continuous uniform distribution in the interval $(0,\theta)$. Therefore, the likelihood is given by:
$f(x|\theta) = \frac{1}{\theta}I(x\leq\theta)$, where $I$ is an indicator function.
But what is the Jeffreys prior for the parameter $\theta$? In particular, how to handle the indicator function when calculating the Fisher information?
Best Answer
I think the other answer is wrong, so will give a detailed development here. First, let $X_1, \dotsc, X_n$ be iid uniform on the interval $(0,\theta)$. Then the likelihood function can be written as $$ L(\theta)= \theta^{-n} \cdot \mathbb{1}(\theta \ge T) $$ where $T=\max(X_1, \dotsc, X_n)$ is the sufficient statistic for $\theta$. The log likelihood then can be written $$ l(\theta)=\log L(\theta)= -n \log \theta+ \begin{cases} 0 &,\theta\ge T \\ -\infty &, \theta<T \end{cases} $$ and its first derivative (where it exists) can be written $$ -n/\theta $$ with expectation equal to $ -n/\theta \not= 0$, so **we cannot calculate the Fisher information via the expectation of the second derivative, since that equality depends on equality to zero above. If we nevertheless do that, we end up with Fisher information as $-n/\theta^2$, which is negative so of course impossible.
Then, using directly the definition of Fisher information, see Wikipedia: Fisher information, we get $$\DeclareMathOperator{\E}{\mathbb{E}} I(\theta)=\E_{\theta}\left\{ \left[\frac{\partial}{\partial\theta}\log f(x;\theta)\right]^2\right\} = \int_0^\theta [-n/\theta]^2 (1/\theta)\; dx = (n/\theta)^2 $$ Then the Jeffrey's uninformative prior is proportional to its squareroot, that is, $$ \pi(\theta) \propto 1/\theta, \quad \theta>0 $$ which is an improper prior. But when $n\ge 2$ we get a proper posterior, which is Pareto with parameters of scale $x_m = L$ and shape $\alpha = n$. The density is given by $$ \pi(\theta \,|\, T) = \frac{n}{\theta} \left( \frac{T}{\theta}\right)^{n} $$ for $\theta \ge T$ and $n\ge 2$.
Note that the Jeffrey's prior can be seen as a degenerate Pareto conjugate prior with parameters $x_m = 0, \, \alpha = 0$.