I need to find distribution of $U=(Y-X_1\hat{\beta_1})^T(Y-X_1\hat{\beta_1})$.

Suppose a linear model $Y= X\beta +\varepsilon$ where $\varepsilon \sim N(0,\sigma ^2I)$. Write $X=[X_1;X_2]$ where $X_1$ are the first $p_1$ columns of $X$ nad $X_2$ are the last $p_2$ columns. Similary split $\beta ^T =(\beta_1^T; \beta_2^T)$.

I computed:

$\operatorname E(Y - X_1\hat{\beta_1}) = X\beta - X_1\beta _1$

$\operatorname{var}(Y - X_1\hat{\beta_1}) = \sigma ^2(I-H_1)$

where $(I-H_1)$ is idemotent matrix. Then I computed $(I-H_1)(X\beta - X_1\beta _1) =0$

Here I stopped and I do not know what I need to do next.

Thank you for any help.

  • $\begingroup$ It depends on how the error terms are distributed, but I suspect you'll find the model's assumptions are such that $Y-X_1\hat{\beta}_1$ has a mean-zero multivariate Gaussian distribution, in which case $U$ has this distribution: en.wikipedia.org/wiki/Wishart_distribution $\endgroup$ – J.G. Mar 8 '18 at 10:17

Let $\beta_1 \in \mathbb{R}^{p_1}$ and $Y\in \mathbb{R}^n$, and recall that if $X_1,...,X_2 \sim \mathcal{N}(\mu, \sigma^2)$ then $\sum_{i=1}^n \sigma^{-2}(X_i - \mu)^2 \sim \chi^2_{(n)}$. So, $$ U = (Y - X_1 \hat{\beta}_1)'( Y - X_1 \hat{\beta}_1) =e_1'e_1 = ||e||_2^2 \sim \sigma^2\chi^2_{(n-p_1)}. $$

| cite | improve this answer | |

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.