Maximum likelihood estimator of mean difference

[safina]

Homework Statement

A sample of size n_{1} is to be drawn from a normal population with mean \mu_{1} and variance \sigma^{2}_{1}. A second sample of size n_{2} is to be drawn from a normal population with mean \mu_{2} and variance \sigma^{2}_{2}. What is the maximum likelihood estimator of \theta = \mu_{1} - \mu_{2}?

If we assume that the total sample size n = n_{1} + n_{2} is fixed, how should the n observations be divided between the two populations in order to minimize the variance of the maximum likelihood estimator of \theta?

Homework Equations

The Attempt at a Solution

 

[lanedance]

so if we denote the samples from distribution 1, xi, and the samples from distribution 2, yj, then let call the samples
\textbf{x} = (x_1,.., x_i,..)
\textbf{y} = (x_1,.., y_j,..)

i'd start by trying to find the distribution likelihood of the data given the sample parameters

L(\textbf{x}, \textbf{y} | \mu_1, \sigma_1, \mu_2, \sigma_2)

then consider differntiating ln(L) w.r.t. theta

(I haven't tried this, but its how i'd try and get started)

 

[lanedance]

also are any of the parameters known beforehand? if all are unknown as they are not independent you could consider the set \left{\theta,\sigma_1,\mu_2, \sigma_2 \right}

 

[lanedance]

And one more let, t, m1, m2 be the estimators for theta, mu1 and mu2

Due to the independence, it shouldn't be too hard to convince yourself the MLE for t is
t=m1+m2

Now you should be able to come up with the sampling distribution for m1 and m2
P(m1,m2|mu1,mu2,sig1,sig2) and use that to find a sampling distribution for t, of which the variance should be apparent. Then minimize wrt the constraint

 

[safina]

also are any of the parameters known beforehand? if all are unknown as they are not independent you could consider the set \left{\theta,\sigma_1,\mu_2, \sigma_2 \right}

None of the parameters are known beforehand.

I have a feeling that the MLE for t is m1-m2, but could you help me more what will be the form of the likelihood function?