Fisher Information of $n$ for $\mathrm{Binomial}(n,p)$ / Fisher information does not exist for distributions with parameter-dependent supports.

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Why is it that "Fisher information does not exist for distributions with parameter-dependent supports"? Why is Fisher Information of $n$ for $\mathrm{Binomial}(n,p)$ an ill-defined question? I guess this shows I actually don't really understand what Fisher information means intuitively. I found the comment here: What is the Fisher information for a Uniform distribution?

score 1 · Accepted Answer · answered Dec 10 '19 at 00:12

Let $X$ be a real random variable with density of the form $$f(x;\theta)=g(x;\theta)I(x \leq \theta_2),$$ where $\theta=(\theta_1,\theta_2)\in R^2$ is the unknown parameter and $I$ is the indicator function. The likelihood in this case is (check) $$L(\theta;x) = \left(\prod_{i=1}^ng(x_i;\theta)\right)I_{[x_{(n)},\infty)}(\theta_2).$$ The important observation is that $L$ is not derivable in $(\theta_1,x_{(n)})$ for any $\theta_1$! So you can not define (at least directly) the Fisher information.

Fisher Information of $n$ for $\mathrm{Binomial}(n,p)$ / Fisher information does not exist for distributions with parameter-dependent supports.

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