As mentioned in the title, I want to derive the Cramer-Rao Lower bound from the Hammersly-Chapman-Robbins lower bound for the variance of a statistic $T$. The statement for the H-C-R lower bound is the following,
Let $\mathbf{X} \sim f_{\theta}(.)$ where $\theta \in \Theta \subseteq \mathbb{R}^k.$ Suppose $T(\mathbf{X})$ is an unbiased estimator of $\tau(\theta)$ where $\tau \colon \Theta \to \mathbb{R}$. Then we have, \begin{equation*} \text{Var}_{\theta}(T) \ge \displaystyle \sup_{\Delta \in \mathcal{H}_{\theta}}\, \displaystyle \frac{[\tau(\theta + \Delta) - \tau(\theta)]^2}{\mathbb{E}_{\theta}\left(\frac{f_{\theta + \Delta}}{f_{\theta}} - 1\right)^2} \end{equation*} where $\mathcal{H}_{\theta} = \{\alpha \in \Theta \colon \text{ support of } f \text{ at } \theta + \alpha \subseteq \text{ support of } f \text{ at } \theta\}$
Now when $k = 1$ and the regularity conditions hold, taking $\Delta \to 0$ gives the following inequality, \begin{equation*} \text{Var}_{\theta}(T) \ge \displaystyle \frac{[\tau'(\theta)]^2}{\mathbb{E}_{\theta} \left( \frac{\partial }{\partial \theta} \log f_{\theta}(\mathbf{X}) \right)^2} \end{equation*} which is exactly the C-R inequality for univariate case.
However, I want to derive the general form of C-R inequality from the H-C-R bound, i.e. when $k > 1$. But, I have not been able to do it. Though, I was able to figure out that we would have to use $\mathbf{0} \in \mathbb{R}^k$ instead of $0$ and $|\Delta|$ to obtain the derivatives, which was obvious anyways, I couldn't get to any expression remotely similar to the C-R inequality. One of the difficulty arises while dealing with the squares. Since for the univariate case, we were able to take the limit inside and as a result got the square of the derivative. While, for the latter case, we cannot take the limit inside, because the derviate in this case would be a vector and we will have the expression containg the square of a vector which is absurd.
I want to know how to derive the C-R inequality in the latter case?