How do you establish the regularity conditions for the Cramér–Rao lower bound for the sample variance estimator?

Question

Let $X_1,X_2,\ldots,X_n \sim \text{IID } f(\theta)$ be a random sample from a distribution with parameter $\theta$ and let $S^2(\mathbf{x}_n) \equiv \frac{1}{n-1} \sum_{i=1}^n (x_i -\bar{x}_n)^2$ denote the sample variance. I want to check the regularity conditions for the Cramér–Rao lower bound, namely:

$$\begin{align} &(1) & & \mathbb{V}_\theta(S^2(\mathbf{X}_n))< \infty, \\[10pt] &(2) & & \frac{\partial}{\partial \theta} \int S^2 (\mathbf{x}_n) f(\mathbf{x}_n | \theta) \ dx = \int S^2(\mathbf{x}_n) \frac{\partial f}{\partial \theta} (\mathbf{x}_n | \theta) \ dx. \\[6pt] \end{align}$$

I would say that $(1)$ is obvious, since $S^2$ is finite, but I do not know what to do with $(2)$. Could you help me?

Answer 1

Actually, condition $(1)$ is not satisfied unless you impose an important moment condition on the distribution of your random variables. The variance of the sample variance for IID random variables has a known form, and it is finite if and only if the underlying distribution has finite kurtosis. So condition $(1)$ is only satisfied if this is the case.

Condition $(2)$ is a condition involving the allowability of bringing the derivative operator inside the integral. The general rule for derivatives of integrals is given by Leibniz integral rule, and the regularity condition holds in the case where the support of the underlying distribution does not depend on the parameter $\theta$.