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How to find the OLS estimator of variance of error
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Given the Linear Regression model $y=Xbeta+epsilon$, where $epsilon sim D(0_n,sigma^2 I_n)$ and $D$ is some distribution. How to find the OLS estimator of $sigma^2$.
I know that the sum of least-squares residuals has a distribution equal to $sigma^2$ times $chi^2_n-1$. So I can therefore determine the unbiased estimator of $sigma^2$. But how to prove that sum of least-squares residuals follows such distribution. Also anyone with new way to find it are welcome.
regression least-squares
asked 6 hours ago
Sarath
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up vote
1
down vote
favorite
Given the Linear Regression model $y=Xbeta+epsilon$, where $epsilon sim D(0_n,sigma^2 I_n)$ and $D$ is some distribution. How to find the OLS estimator of $sigma^2$.
I know that the sum of least-squares residuals has a distribution equal to $sigma^2$ times $chi^2_n-1$. So I can therefore determine the unbiased estimator of $sigma^2$. But how to prove that sum of least-squares residuals follows such distribution. Also anyone with new way to find it are welcome.
regression least-squares
asked 6 hours ago
Sarath
62
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Sarath is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
In $D(0_n,sigma^2 I_n)$, if $0_n$ is the mean and $sigma^2$ = variance, which kind of distributions meet this condition, apart from normal?
– a_statistician
1 hour ago
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up vote
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down vote
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up vote
1
down vote
favorite
Given the Linear Regression model $y=Xbeta+epsilon$, where $epsilon sim D(0_n,sigma^2 I_n)$ and $D$ is some distribution. How to find the OLS estimator of $sigma^2$.
I know that the sum of least-squares residuals has a distribution equal to $sigma^2$ times $chi^2_n-1$. So I can therefore determine the unbiased estimator of $sigma^2$. But how to prove that sum of least-squares residuals follows such distribution. Also anyone with new way to find it are welcome.
regression least-squares
asked 6 hours ago
Sarath
62
New contributor
Sarath is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Given the Linear Regression model $y=Xbeta+epsilon$, where $epsilon sim D(0_n,sigma^2 I_n)$ and $D$ is some distribution. How to find the OLS estimator of $sigma^2$.
I know that the sum of least-squares residuals has a distribution equal to $sigma^2$ times $chi^2_n-1$. So I can therefore determine the unbiased estimator of $sigma^2$. But how to prove that sum of least-squares residuals follows such distribution. Also anyone with new way to find it are welcome.
regression least-squares
regression least-squares
asked 6 hours ago
Sarath
62
New contributor
Sarath is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 6 hours ago
Sarath
62
New contributor
Sarath is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 5 hours ago
asked 6 hours ago
Sarath
62
New contributor
Sarath is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 6 hours ago
Sarath
62
asked 6 hours ago
Sarath
62
62
New contributor
Sarath is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Sarath is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Sarath is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
In $D(0_n,sigma^2 I_n)$, if $0_n$ is the mean and $sigma^2$ = variance, which kind of distributions meet this condition, apart from normal?
– a_statistician
1 hour ago
add a comment |Â
In $D(0_n,sigma^2 I_n)$, if $0_n$ is the mean and $sigma^2$ = variance, which kind of distributions meet this condition, apart from normal?
– a_statistician
1 hour ago
In $D(0_n,sigma^2 I_n)$, if $0_n$ is the mean and $sigma^2$ = variance, which kind of distributions meet this condition, apart from normal?
– a_statistician
1 hour ago
In $D(0_n,sigma^2 I_n)$, if $0_n$ is the mean and $sigma^2$ = variance, which kind of distributions meet this condition, apart from normal?
– a_statistician
1 hour ago
add a comment |Â
1 Answer
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The OLS (Ordinary Least Squares) estimate does not depend on the distribution D, so for any distribution you can use the exact same tools as the for the normal distribution.
This just gives the OLS estimates of the parameters, it does not justify any tests or other inference that could depend on your distribution D (though the Central Limit Theorem holds for regression and for large enough sample sizes (how big depends on how non-normal D is) the normal based tests and inference will still be approximately correct.
If you want Maximum Likelihood estimation instead of OLS, then this will depend on D). The normal distribution has the advantage that OLS gives the Maximum Likelihood answer as well.
answered 5 hours ago
Greg Snow
38.8k157121
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
The OLS (Ordinary Least Squares) estimate does not depend on the distribution D, so for any distribution you can use the exact same tools as the for the normal distribution.
This just gives the OLS estimates of the parameters, it does not justify any tests or other inference that could depend on your distribution D (though the Central Limit Theorem holds for regression and for large enough sample sizes (how big depends on how non-normal D is) the normal based tests and inference will still be approximately correct.
If you want Maximum Likelihood estimation instead of OLS, then this will depend on D). The normal distribution has the advantage that OLS gives the Maximum Likelihood answer as well.
answered 5 hours ago
Greg Snow
38.8k157121
add a comment |Â
up vote
3
down vote
The OLS (Ordinary Least Squares) estimate does not depend on the distribution D, so for any distribution you can use the exact same tools as the for the normal distribution.
This just gives the OLS estimates of the parameters, it does not justify any tests or other inference that could depend on your distribution D (though the Central Limit Theorem holds for regression and for large enough sample sizes (how big depends on how non-normal D is) the normal based tests and inference will still be approximately correct.
If you want Maximum Likelihood estimation instead of OLS, then this will depend on D). The normal distribution has the advantage that OLS gives the Maximum Likelihood answer as well.
answered 5 hours ago
Greg Snow
38.8k157121
add a comment |Â
up vote
3
down vote
up vote
3
down vote
The OLS (Ordinary Least Squares) estimate does not depend on the distribution D, so for any distribution you can use the exact same tools as the for the normal distribution.
This just gives the OLS estimates of the parameters, it does not justify any tests or other inference that could depend on your distribution D (though the Central Limit Theorem holds for regression and for large enough sample sizes (how big depends on how non-normal D is) the normal based tests and inference will still be approximately correct.
If you want Maximum Likelihood estimation instead of OLS, then this will depend on D). The normal distribution has the advantage that OLS gives the Maximum Likelihood answer as well.
answered 5 hours ago
Greg Snow
38.8k157121
The OLS (Ordinary Least Squares) estimate does not depend on the distribution D, so for any distribution you can use the exact same tools as the for the normal distribution.
This just gives the OLS estimates of the parameters, it does not justify any tests or other inference that could depend on your distribution D (though the Central Limit Theorem holds for regression and for large enough sample sizes (how big depends on how non-normal D is) the normal based tests and inference will still be approximately correct.
If you want Maximum Likelihood estimation instead of OLS, then this will depend on D). The normal distribution has the advantage that OLS gives the Maximum Likelihood answer as well.
answered 5 hours ago
Greg Snow
38.8k157121
answered 5 hours ago
Greg Snow
38.8k157121
answered 5 hours ago
Greg Snow
38.8k157121
answered 5 hours ago
Greg Snow
38.8k157121
38.8k157121
add a comment |Â
add a comment |Â
Sarath is a new contributor. Be nice, and check out our Code of Conduct.
Sarath is a new contributor. Be nice, and check out our Code of Conduct.
Sarath is a new contributor. Be nice, and check out our Code of Conduct.
Sarath is a new contributor. Be nice, and check out our Code of Conduct.
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In $D(0_n,sigma^2 I_n)$, if $0_n$ is the mean and $sigma^2$ = variance, which kind of distributions meet this condition, apart from normal?
– a_statistician
1 hour ago