Confidence Interval for Maximum Likelihood Estimate

[AngryJames]

This should probably be very simple but I'm confused as to how to calculate an estimate of the variance required to construct a confidence interval for a maximum likelihood estimate.

In the answers to the question I'm looking at it says that the estimate for the variance is given by:

var (p) = -1/ E[(d^2/dp^2) (ln L)]

I hope this notation makes sense. Is anyone able to explain where this comes from please?

 

[Calum]

That's the Cramer-Rao lower bound. If you covered CT3, it's in there - if not, it's probably best to dip into a stats text - Casella & Berger's treatment is not too hard going as I recall.

This is the derivation according to Wikipedia:

http://en.wikipedia.org/wiki/Cramér–Rao_bound#Single-parameter_proof

which may explain where it comes from if you're comfortable with the algebra.

 

[AngryJames]

Thanks Calum

I haven't got the notes for CT3 sadly as through my probability and statistics at University I've gained an exemption. I never covered the Cramer-Rao lower bound though, or at least that I can remember!

I'll take a look through this. It looks spot on what I'm after.

 

[Slumpy]

Thanks Calum

I haven't got the notes for CT3 sadly as through my probability and statistics at University I've gained an exemption. I never covered the Cramer-Rao lower bound though, or at least that I can remember!

I'll take a look through this. It looks spot on what I'm after.

For the exam, if you need it, the formula is also on page 23 of the tables.

 

[Mark Mitchell]

From the point of view of the CT4 exam, you just need to be able to apply the formula for the CRLB on page 23 of the Tables, rather than know the derivation. In case that puts your mind at rest.

 

[AngryJames]

Thanks Slumpy, that's also really helpful.

Thanks Mark. It's always nice when it makes sense where things come from! I won't worry if I don't remember this when it comes to the exam though.