Sept 2007 Q.12 standard error

In part of ii) of this question we are asked to calculate the standard error for theta(hat). I've seen the standard error calculated in chapter 13 (correlation & regression) but don't know how to apply it here.

Can anyone help please?

 

have you read the examiner's reports here for your exam (sept 2007)?

http://www.actuaries.org.uk/students/exams/preparing/exam_papers/2007_papers

It looks like they are using the Cramer Rao lower bound to estimate it.

 

Yeah, I saw the examiners report and couldn't work out what they were doing. Why are they using Cramer Rao?

 

Hi again Sullyfer

EDIT your not even asking bout this arrgh read aussie comment and thought you wanted to know about part (b) EDIT


you really dont like this crlb stuff , we use the crlb because as the name states its a lower bound for the variance of our estimator , and since we have a "large sample" given in the question, we can be fairly confident that the our variance is fairly close to the CRLB so theta is approximately distributed
Normal ( theta(hat) , CRLB )
hope this helps its been awhile since I looked at this so maybe someone else could confirm???

 

Now the real answer the one you've been waiting for or not

this is pretty easy actually you didnt even need to be able to do part (i) to do (ii) your just filling in numbers
first part theta hat is n over m from (i)b so 100/45

second and main part take CRLB formula from part (i)c and sub in for theta hat and n , and take the root cause were asked for s.d. not variance

idea is ,n large ,100 data points ,time enough for the variance to converge in on the CRLB

I hope you noticed the distribution at the start of the question is quite similar to binomial but here p = 1/theta and q = 1-p = 1 - 1/theta
so i knew before i started that theta hat was going to be 1/ p hat
just done MLE on binomial way to many times I suppose

 

i'm really missing something here......
I can't find anywhere in the notes where it explains what standard error is apart from in the correlation & regression chapter.
I haven't done stats before.
Can someone explain to me what it is and how i calculate it.
I can see here that the CRLB is being used, but I don't know why...
and in q.13 sept 2005, part ii) the examiners use the standard error again to calculate a test statistic for the difference between 2 polulation proportions.

I'm confused

 

The standard error of an estimator is the square root of the variance. It is the same as standard deviation, but in the context of an estimator it is called standard error.

The CRLB is being used here because this is a maximum likelihood estimator.

Maximum likelihood estimators are asympotically normally distributed, asymptotically unbiased and asymptotically their variance is given by the CRLB.