Point estimation with MLE and MoM

[MarkSA]

Hi,

My question to answer is: Estimate Lambda in a Poisson distribution if X1, X2, etc. are iid, with the:
a) Method of Moments.
b) Maximum Likelihood Estimator.
Are the estimators unbiased? Are they the best estimators for Lambda?

I'm really having trouble understanding how these two methods work. I just don't see what's going on. I've read so much about them though that I think I have the answers to the questions at least. For Method of moments & MLE, estimator is 1/n * (Sum from i=1 to n of: x[sub:bzzisawh]i[/sub:bzzisawh]) Both unbiased estimators, and they are efficient estimators.

But I really want to understand what's going on here, so I was wondering if someone could explain how each of these is derived? From what i've gathered both can be found from the pmf of poisson: P(X=x)=[Lambda^x * e^(-Lambda)]/x!
I'm not sure how to derive them though.

 

[royhaas]

MOM solves for \(\displaystyle \lambda\) in terms of the mean of the distribution. MLE solves for the value of \(\displaystyle \lambda\) which maximizes the log-likelihood of the (log of) the pmf of a sample. It just so happens they are equal in the Poisson case. This is not always what happens

 

[DrMike]

Not so sure about the method of moments, but I can show you how MLE works.

The idea is : "Given this data, what set of parameters assign this data the highest probability?"

So, if you have X1, X2, X3... Xn drawn independently from a normal distribution N(mu, sig^2), with KNOWN sig and UNKNOWN mu, the probability of getting (within dX1, dX2, dX3.... of) this data is

constant * exp(-(X1-mu)^2/(2*sig^2)) * exp(-(X2-mu)^2/(2*sig^2)) * ... * exp(-(Xn-mu)^2/(2*sig^2)) * dX1 * dX2 * ... * dXn

Now we need to find mu to maximise this, ie, to maximise

-(X1-mu)^2/(2*sig^2)-(X2-mu)^2/(2*sig^2)-...-(Xn-mu)^2/(2*sig^2), that is, to minimise the sum of (Xi-mu)^2. Differentiate, solve, and mu=sum(Xi)/n

If we didn't know sig either, the 'constant' wouldn't be constant any more, and we'd have to minimise a function of two variables mu and sig.