MLE help

[allinorfold]

Hey guys, I've a Linear Models assignment due in 3 weeks. Really appreciate some help in certain area(s).

Here's the first part:
I'll explain the experiment,
You are counting salmon for K days. You have a perfect detector for first n days and an imperfect detector K-n days. Imperfect detects fish with probability p.


Assume:
perfect detector is
used for the first n days and the imperfect detector for the next K-n days and that m1 and m2
salmon are detected during those periods, respectively.

1•.Show carefully that the MLE for (, p) is given by
ˆ=m1/n, pˆ= m2n/m1(K-n), in the case where m2n/m1(K-n) is less than or equal to 1.


2. Find the MLE for (, p) when this latter condition is not
satisfied.


((''''' note: m1 and m2 here are m subscript 1 and m subscript 2 respectively. I’m not sure how to type them properly in ‘Word’.''''''))
and also the squares are obviously ''lambda''.

The next part is a covariance matrix problem. I have the answers for the above and I have worked out a way to get them, but I'm not entirely sure if my ways are correct. I'm presuming, the Liklihood is poisson(lambda) x poisson(plambda).?? Any help would be great, cheers.

[mp83]

You pressumed right! Nothing to be afraid of...

[allinorfold]

Well, that's good news.
However,
to estimate lambda^, I just use the poisson(lambda).
and to estimate p^, I just use the poisson(plambda).

This is what I used and got the answer that the question says. I admit, I worked backwards basically. Joining the two together did not give me the correct answer?
Would I be correct in saying this?

[allinorfold]

Sorry, I should stress that I used

L(lambda) is proportional to lambda^(m1) x e^(-nlambda)

to estimate lambda^.


(and similar for plambda.) replacing m1 with m2, lambda with plambda, and n with (K-n).

This is the Poisson dist for i.i.d. r.v.'s I believe??

[allinorfold]

For the record,
it was distributed poisson(lambda x n) and poisson(lambda x p x (k-n))