# #StackBounty: #estimation #inference #fisher-information #efficiency Deriving C-R inequality from H-C-R bound

### Bounty: 50

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,
$$begin{equation} text{Var}{theta}(T) ge displaystyle sup{Delta in mathcal{H}{theta}}, displaystyle frac{[tau(theta + Delta)]^2}{mathbb{E}{theta}left(frac{f_{theta + Delta}}{f_{theta}} – 1right)^2} end{equation}$$
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?

Get this bounty!!!

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