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Statsy McStatsface

https://www0.maths.ox.ac.uk/system/files/coursematerial/2015/3188/4/sheet3.pdf

Question 5b) here.

So basically I've the MLE of mu (X bar, using independence and finding the joint distribution). Im fairly certain this is correct, but am unsure how to construct a CI. Do you just construct one for Lj or Rj ?

Also, for 5c) I am unsure how to find the joint density function when Lj and Rj are not independent.

Any help appreciated :smile:

@Gregorius
@ghostwalker
Original post by Gome44


Sorry - somewhat beyond my knowledge base.
Original post by Gome44
https://www0.maths.ox.ac.uk/system/files/coursematerial/2015/3188/4/sheet3.pdf

Question 5b) here.

So basically I've the MLE of mu (X bar, using independence and finding the joint distribution). Im fairly certain this is correct, but am unsure how to construct a CI. Do you just construct one for Lj or Rj ?


You are given that the Lj and the Rj are all independent of each other. So the situation is as if you simply had a random sample of 2n eye pressure measurement, where it simply doesn't matter whether the measurement has come from a left eye or a right eye! Your likelihood function with therefore give you an MLE for μ \mu that involves a sum up to 2n.

Also, for 5c) I am unsure how to find the joint density function when Lj and Rj are not independent.


Do you know how to write down the distribution function of a multivariate normal distribution using a variance-covariance matrix to describe the dependence structure?

In this case the covariance between Lj and Rj is easy to write down:

Cov(Lj,Rj)=Cov(Mj+Dj,Mj−Dj)=σ12−σ22 \displaystyle Cov(Lj, Rj) = Cov(Mj + Dj, Mj - Dj) = \sigma_{1}^{2} - \sigma_{2}^{2}
(edited 7 years ago)
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Gome44
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Original post by Gregorius
You are given that the Lj and the Rj are all independent of each other. So the situation is as if you simply had a random sample of 2n eye pressure measurement, where it simply doesn't matter whether the measurement has come from a left eye or a right eye! Your likelihood function with therefore give you an MLE for μ \mu that involves a sum up to 2n.



Do you know how to write down the distribution function of a multivariate normal distribution using a variance-covariance matrix to describe the dependence structure?

In this case the covariance between Lj and Rj is easy to write down:

Cov(Lj,Rj)=Cov(Mj+Dj,Mj−Dj)=σ12−σ22 \displaystyle Cov(Lj, Rj) = Cov(Mj + Dj, Mj - Dj) = \sigma_{1}^{2} - \sigma_{2}^{2}


Thanks :smile:. I did not know about the variance-covariance matrix, and I'm not sure if we are supposed to or not, but will read up on it.

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