Fisher’s lost information

After a post on X validated and a good discussion at work, I came to the conclusion [after many years of sweeping the puzzle under the carpet] that the (a?) Fisher information obtained for the Uniform distribution U(0,θ) as θ⁻¹ is meaningless. Indeed, there are many arguments:

  1. The lack of derivability of the indicator function for x=θ is a non-issue since the derivative is defined almost everywhere.
  2. In many textbooks, the Fisher information θ⁻² is derived from the Fréchet-Darmois-Cramèr-Rao inequality, which does not apply for the Uniform U(0,θ) distribution.
  3. One connected argument for the expression of the Fisher information as the expectation of the squared score is that it is the variance of the score, since its expectation is zero. Except that it is not zero for the Uniform U(0,θ) distribution.
  4. For the same reason, the opposite of the second derivative of the log-likelihood is not equal to the expectation of the squared score. It is actually -θ⁻²!
  5. Looking at the Taylor expansion justification of the (observed) Fisher information, expanding the log-likelihood around the maximum likelihood estimator does not work since the maximum likelihood estimator does not cancel the score.
  6. When computing the Fisher information for an n-sample rather than a 1-sample, the information is n²θ⁻², rather than nθ⁻².
  7. Since the speed of convergence of the maximum likelihood estimator is of order n⁻², the central limit theorem does not apply and the limiting variance of the maximum likelihood estimator is not the Fisher information.