# MLE and Regression. Insights

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#1
Hiya!

I wanted to know if anyone knows any sites for understanding regression (multiple and simple). I'm finding it all very confusing. I have a book by Hogg & Tanis, but am still a little confused by Regression and MLE, as well as ANOVA. That whole side of statistics is rather confusing for me.

Also, I don't understand how:

1) The arithmetic mean is the MLE of the unknown mean/mew of the normal pdf. How do I prove that or is there proof of that anywhere?

2) And why the arithmetic mean could be a sufficient statistic for the unknown mean and nor do I understand how it's distribution could be that of X(bar)~N(mew, sigma squared/n). Also, if i'm not mistaken, the pdf could be used to show that the arithmetic mean is the MLE of unknown mew, but is there a proof of that? If not, could someone show me how that could be the case?

I have an exam at the beginning of June, so need to get the understanding in there pretty quickly. Also! Hypothesis testing has me confused and how that fits in to all this regression stuff.
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6 years ago
#2
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6 years ago
#3
(Original post by Minni04)
1) The arithmetic mean is the MLE of the unknown mean/mu of the normal pdf. How do I prove that or is there proof of that anywhere?

2) And why the arithmetic mean could be a sufficient statistic for the unknown mean and nor do I understand how it's distribution could be that of X(bar)~N(mu, sigma squared/n). Also, if i'm not mistaken, the pdf could be used to show that the arithmetic mean is the MLE of unknown mu, but is there a proof of that? If not, could someone show me how that could be the case?
It took me a moment to realise that by "mew" you meant "mu"… it distracted me so much that I amended it in my quotation of you.

Yes. Have you found the MLE of the unknown mean? (by finding the likelihood or log-likelihood, and differentiating to maximise it - this is how the PDF can be used to show that the arithmetic mean is the MLE of unknown mu)

A statistic is sufficient if the PDF factorises into a term with the statistic, and a term without. What is the PDF?

Do you know how to find the distribution of the sum of normal random variables, and how to find the distribution of 1/n times that sum?
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#4
(Original post by Smaug123)
It took me a moment to realise that by "mew" you meant "mu"… it distracted me so much that I amended it in my quotation of you.

Yes. Have you found the MLE of the unknown mean? (by finding the likelihood or log-likelihood, and differentiating to maximise it - this is how the PDF can be used to show that the arithmetic mean is the MLE of unknown mu)

A statistic is sufficient if the PDF factorises into a term with the statistic, and a term without. What is the PDF?

Do you know how to find the distribution of the sum of normal random variables, and how to find the distribution of 1/n times that sum?
Hahahah i'm so sorry.. I've never ever seen it written down, so thought I'd spell it out phonetically.

I've managed to find out the MLE of the unknown mean.. At least I think I did. It follows a normal distribution.

Could you just explain the log part again please? How would i find the log likelihood function?

Is the factorization theorem needed for the sufficient statistic then?

"Do you know how to find the distribution of the sum of normal random variables, and how to find the distribution of 1/n times that sum" Unfortunately no, I don't
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6 years ago
#5
(Original post by Minni04)
I've managed to find out the MLE of the unknown mean.. At least I think I did. It follows a normal distribution.
Yes, it does.

Could you just explain the log part again please? How would i find the log likelihood function?
The log-likelihood is the log of the likelihood. It's usually easier to calculate with the log-likelihood (eg. for differentiation purposes) - because log is strictly increasing, it preserves turning points and so on, so maximising log-likelihood is equivalent to maximising likelihood. Presumably you differentiated something in order to find the MLE?

Is the factorization theorem needed for the sufficient statistic then?
It's not *needed*, but it makes it a lot easier for finding sufficient statistics/proving that statistics are sufficient…

"Do you know how to find the distribution of the sum of normal random variables, and how to find the distribution of 1/n times that sum" Unfortunately no, I don't
The sum of random normal iid variables is normal, for instance?
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