# Different formulas for the Fisher Information, for use in Cramér-Rao lower bound. Correct? Assumptions?

I am taking a course in Statistical Inference. We have to calculate the Cramér-Rao lower bound to determine if different statistics are efficient or not.

I have read that as long as we have an unbiased estimator, the CRLB is just the reciprocal of the Fisher Information. Using this, we were handed a lot of different formulas for the Fisher Information in an exercise session. This was however not covered in too much depth.

What I have so far is the following:

Let $$f(\mathbf x | \theta)$$ be the joint distribution function for the observed sample, then the Fisher Information is

$$I(\theta) = \text{E} \left[ \left( \frac{d}{d \theta} \log f(\mathbf x | \theta) \right)^2 \right] = - \text{E} \left[ \frac{d^2}{d \theta ^2} \log f(\mathbf x | \theta) \right].$$

If the observations are i.i.d. the Fisher Information can also be written as $$I(\theta) = n \text{E} \left[ \left( \frac{d}{d \theta} \log f(x | \theta) \right)^2 \right] = -n \text{E} \left[ \frac{d^2}{d \theta^2} \log f(x | \theta) \right],$$ where $$f(x|\theta)$$ is the pdf for each observed variable.

My questions are:

• Are these formulas correct?
• Are there any other assumptions besides those mentioned required, and are the mentioned assumptions correct?
• Are these variants sufficient given that we will only consider unbiased estimators?
• With some regularity conditions, the definition is the very first of your formulas with $\mathbf x$ usually replaced by the random variable $\mathbf X$. The one with the second derivative does not always hold. For full details, you can see Elements of Large-Sample Theory by Lehmann (page 456 onwards). – StubbornAtom Jan 8 at 8:43