Fisher Information Matrix: Equivalent Expressions

[weetabixharry]

I don't understand the following step regarding the $(i,j)^{th}$ element of the Fisher Information Matrix, $\textbf{J}$:
$$J_{ij}\triangleq\mathcal{E}\left\{ \frac{\partial}{\partial\theta_{i}}L_{\textbf{x}}(\textbf{θ})\frac{\partial}{\partial\theta_{j}}<br /> <br /> L_{\mathbf{x}}(\textbf{θ})\right\}<br /> \\<br /> =-\mathcal{E}\left\{ \frac{\partial^{2}}{\partial\theta_{i} \partial \theta_{j}}L_{\textbf{x}}(\textbf{θ})\right\}$$

which is given in (Eq. 8.26, on p. 926 of) "Optimum Array Processing" by Harry van Trees. I don't know if the details matter, but $L_{\textbf{x}}$ is the log-likelihood function and he is looking at the problem of estimating the non-random real vector, $\textbf{θ}$, from discrete observations of a complex Gaussian random vector, $\textbf{x}$.

Am I missing something obvious? I'm not very sharp on partial derivatives.

 

[DrDu]

L_X is minus the log of the pdf. Write down the explicit expression for the expectation values in terms of L_X and its derivatives and use partial integration.

 

[weetabixharry]

Write down the explicit expression for the expectation values in terms of L_X and its derivatives and use partial integration.

The expressions are rather intimidating functions of complex matrices. I don't think I want to try partial integration. I tried just evaluating the derivatives to see if they would come out the same, but I ran out of paper before I had even scratched the surface.

If the result is specific to this problem, then I would be willing to take it on face value. It's just frustrating that I keep seeing the result stated without proof. It's as though it's too obvious to warrant a formal proof.

 

[DrDu]

The problem is that I am too lazy to tex something which can be found in any text on statistics or google, e.g.:

http://mark.reid.name/iem/fisher-information-and-log-likelihood.html

 

[weetabixharry]

The problem is that I am too l

I see.

http://mark.reid.name/iem/fisher-information-and-log-likelihood.html

Thanks - this was useful; its author claims to have spent 6 months pondering this problem, so I'm glad I'm not the only one to have had difficulty finding the solution.

 

[DrDu]

6 month is completely inacceptable. As I said, this topic is contained in every book on introductory statistics.
Elsewise it is a good idea to search for some lecture notes containing the problem: I used "fisher information lecture notes" and almost every lecture note contained a proof of the statement, e.g. the first one i got:
http://ocw.mit.edu/courses/mathemat...ications-fall-2006/lecture-notes/lecture3.pdf

 

[weetabixharry]

this topic is contained in every book on introductory statistics.

This is obviously incorrect.

Elsewise it is a good idea to search for some lecture notes containing the problem

This is no longer necessary. The topic is resolved.