2
$\begingroup$

I am fitting my data to a multivariate linear regression $Y = BX + \Xi$, where the response is bivariate $Y\in R^{n\times 2}$, and the predictor is uni-variate but elevated to the projective plane to account for the intercept $X\in R^{n\times 2}$.

Now, finding the best fit reduces to $\hat B = (X^T X)^{-1}X^T Y$.

But I am interested in finding a $0.7$ confidence region around $\hat B$. How do I do that?

$\endgroup$
0
$\begingroup$

Looking at https://en.wikipedia.org/wiki/Simple_linear_regression :

This t-value has a Student's t-distribution with $n-2$ degrees of freedom. Using it we can construct a confidence interval for $\beta$:

$$ \beta \in \left[\widehat\beta - s_{\widehat\beta} t^*_{n - 2},\ \widehat\beta + s_{\widehat\beta} t^*_{n - 2}\right] $$

at confidence level $1-\gamma$, where $t^*_{n - 2}$ is the $(1-\frac{\gamma}{2})$-th quantile of the $t_{n−2}$ distribution.

$\endgroup$
1
  • $\begingroup$ So this would work if the regression I was doing was 1D, but I want to extend it to the case where the response variable is 2D $\endgroup$ – Jsevillamol Jan 25 '20 at 20:51
0
$\begingroup$

Bayesian linear regression can provide an estimate for the confidence region for a linear regression estimate.

$\endgroup$
1
  • $\begingroup$ Can you walk me through the math? What prior do I take? How do I update it efficiently? What is the resulting confidence region based on the posterior? $\endgroup$ – Jsevillamol Mar 24 '20 at 15:08

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.