Regression least squares (boring question? or more to it)

[GreenGoblin]

yi = a + bxi + ei is the simple liner regression model as per is usual

"state the assumptions on the errors ei to justify a least squares fit"

? So is this just that E(ei)=0, i can't see what else is a 'must' for this? what about that they are normally distrubited? i know the properties of the errors but what does it mean state the assumptions?

"obtain the least squars estimators a* and b*"

right but there is no data set? so does this just mean give the formulae for the estimators? is this just simple write it down book work or is there something to do here? do you think an examiner must give full mark if you say just a* = yBAR - b*xBAR, b* = Sxy/Sxx?. I don't know what else they can be asking. i have no official solution to it.
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"suggest an unbiased estimato of a + 10b".

now this one just annoys me the most since we have E(a*)=a and E(b*) = b, with the linearity of exectation, this is just going to be a* + 10b*? so there really is nothing to do in the whole question?

i am just lost if there is some calculations to actually do here since this is a pretty valuable question on exam paper and there appears to be no work to do..

 

[davemk]

Regarding the assumptions, aren't they that the errors should be normally distributed, have constant variance (homoscedasticity) and they should be independant?

As far as I know (and stats isn't really my forte), the assumptions need to be met in order for a linear regression to be valid.

 

[GreenGoblin]

Can someone help/confirm my answers for these questions?

The assumptions on the errors are just the expectation of 0, variance of sigma^2, normally distributed, independenced

the unbiased estimator of a + 10b is JUST a* + 10b* (since Expectation of the estimators is just the estimators themselves, 10 is just a constant.. a nd expectation is linear.. this is a bum question really? are they just testing that you know these basic ideas?)

"obtain the least squares estimators"
do you think i can just say sxy/sxx for b* or do i give the formula? is this all they want and a* = yBAR - b*xBAR

thanks dave mk.!

 

[CaptainBlack]

There is no minimal set of assumptions needed to justify the use of least squares fitting, but you will not get any marks for that in an exam answer. For the model to be coherent you assume that \(E(\varepsilon_i)=0\), but that is an assumption of a linear model.

The normal equation work, and give a least squared fit line independent of any assumptions about the error distributions

That the \(\varepsilon_i\) are homoeostatic independent and normally distributed allow the use of additional theory that tells you about the distribution of residual, goodness of the fit, and optimality etc.

CB

 

[blue_raver22]

As a scientist, it is important to understand the assumptions and methods behind statistical models such as the simple linear regression model. In this case, the assumption for the errors ei is that they are independent and identically distributed with a mean of 0 (E(ei)=0). This means that the errors are not influenced by any other variables and have an average value of 0. Additionally, it is often assumed that the errors are normally distributed, but this is not a "must" assumption.

To obtain the least squares estimators a* and b*, you would need a data set to work with. These estimators are calculated using the formulae you mentioned, but they require actual values for the variables in the model. Without a data set, it is not possible to calculate these estimators.

For the last question, it seems that the examiner is asking for an unbiased estimator of a + 10b, which would involve substituting the estimators a* and b* into the equation. This would result in a* + 10b*, as you mentioned.

It is important to understand the concepts and assumptions behind statistical models, but it is also important to have a data set in order to perform any calculations. Without a data set, it is not possible to fully answer these questions.