Timeline for Least squares regression with variance-covariance matrix of observations
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
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| Mar 14, 2018 at 23:01 | vote | accept | Mathieu | ||
| Mar 14, 2018 at 21:38 | history | edited | amoeba |
edited tags
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| S Mar 14, 2018 at 21:34 | history | edited | gung - Reinstate Monica | CC BY-SA 3.0 |
format math notation in latex
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| S Mar 14, 2018 at 21:34 | history | suggested | semibruin | CC BY-SA 3.0 |
format math notation in latex
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| Mar 14, 2018 at 21:32 | review | Suggested edits | |||
| S Mar 14, 2018 at 21:34 | |||||
| Mar 14, 2018 at 20:59 | comment | added | jld | sure, you're very welcome | |
| Mar 14, 2018 at 20:59 | answer | added | jld | timeline score: 4 | |
| Mar 14, 2018 at 20:47 | comment | added | Mathieu | That's what I suspected by analogy with the homoscedastic case, but I needed confirmation. Is there any way for you to format this as an answer, so that I can accept it? Thanks again. | |
| Mar 14, 2018 at 20:39 | comment | added | jld | when $\Omega$ is known and not estimated, the GLS estimator $\hat \beta$ is of the form $MY$ for some known and nonrandom matrix $M$, so it'd just be $Var(\hat \beta) = M Var(Y) M^T = M \Omega M^T$, and then plug in $M = (X^T \Omega^{-1} X)^{-1}X^T \Omega^{-1}$ to get $Var(\hat \beta) = (X^T \Omega^{-1} X)^{-1}$ and note how this is a direct extension of the variance of $\hat \beta$ when the errors are homoscedastic | |
| Mar 14, 2018 at 20:35 | comment | added | Mathieu | Thanks, that was very helpful. I'm still unsure how to compute the var-cov matrix of the best-fit parameters. A quick Wikipedia search was unsuccessful. Can you please point me in the right direction? | |
| Mar 14, 2018 at 20:26 | review | Close votes | |||
| Mar 14, 2018 at 23:03 | |||||
| Mar 14, 2018 at 20:07 | comment | added | jld | Possible duplicate of How to do regression with known correlations among the errors? | |
| Mar 14, 2018 at 19:54 | comment | added | Mathieu | If you mean $Y=\beta X+\epsilon$ in vector space (thus $\beta=(a,b)$), yes. And yes, I can assume that my errors are gaussian. | |
| Mar 14, 2018 at 19:40 | comment | added | jld | would it be fair to say you have something of the form $Y = X\beta + \varepsilon$ where $\varepsilon \sim \mathcal N(0, \Omega)$ with $\Omega$ known but not diagonal? Or are the errors both dependent and non-gaussian? At the very least, regardless of the actual distribution of $\varepsilon$, are you able to say $E(\varepsilon) = 0$ and $Var(\varepsilon) = \Omega$ with $\Omega$ known? | |
| Mar 14, 2018 at 19:33 | review | First posts | |||
| Mar 14, 2018 at 21:33 | |||||
| Mar 14, 2018 at 19:30 | history | asked | Mathieu | CC BY-SA 3.0 |