Maximum likelihood estimator of mean difference

In summary, the conversation discusses the problem of finding the maximum likelihood estimator of the difference between two means from two normal populations with fixed sample sizes. The process involves finding the distribution likelihood of the data given the sample parameters and differentiating it with respect to theta. The MLE for theta is found to be m1-m2, and the sampling distribution for t can be used to minimize its variance. None of the parameters are known beforehand.
  • #1
safina
28
0

Homework Statement


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

If we assume that the total sample size n = n[tex]_{1}[/tex] + n[tex]_{2}[/tex] 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 [tex]\theta[/tex]?


Homework Equations





The Attempt at a Solution

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

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

[tex] L(\textbf{x}, \textbf{y} | \mu_1, \sigma_1, \mu_2, \sigma_2) [/tex]

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

(I haven't tried this, but its how i'd try and get started)
 
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  • #3
also are any of the parameters known beforehand? if all are unknown as they are not independent you could consider the set [itex] \left{\theta,\sigma_1,\mu_2, \sigma_2 \right}[/itex]
 
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  • #4
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
 
  • #5
lanedance said:
also are any of the parameters known beforehand? if all are unknown as they are not independent you could consider the set [itex] \left{\theta,\sigma_1,\mu_2, \sigma_2 \right}[/itex]
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?
 

1. What is a maximum likelihood estimator (MLE) of mean difference?

A maximum likelihood estimator of mean difference is a statistical method used to estimate the difference between the means of two populations based on a sample from each population. It uses the likelihood function to find the values of the mean difference that are most likely to have produced the observed data.

2. How does MLE of mean difference differ from other methods of estimating mean difference?

MLE of mean difference differs from other methods such as the t-test or ANOVA in that it does not assume a specific distribution for the data. Instead, it uses the likelihood function to find the values of the mean difference that are most likely to have produced the observed data.

3. What are the advantages of MLE of mean difference?

The advantages of MLE of mean difference include its flexibility in not assuming a specific distribution for the data, its ability to handle missing data, and its robustness to outliers.

4. How is the MLE of mean difference calculated?

The MLE of mean difference is calculated by finding the value of the mean difference that maximizes the likelihood function. This can be done using mathematical optimization techniques or by using software packages designed for statistical analysis.

5. How can the accuracy of MLE of mean difference be evaluated?

The accuracy of MLE of mean difference can be evaluated by comparing it to other methods of estimating mean difference, such as the t-test or ANOVA, on simulated or real data sets. Additionally, the standard error of the MLE can be calculated and used to construct confidence intervals for the estimated mean difference.

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