Sampling distribution of the regression estimators
Theorem taught in Linear regression, part of Statistical inference
Sampling distribution of the regression estimators
| Type | Theorem |
| Taught in | Linear regression |
| Part | Statistical inference |
Under simple linear regression,
\widehat\beta_1\sim\operatorname{N}\Bigl(\beta_1,\ \frac{\sigma^2}{S_{xx}}\Bigr), \qquad \widehat\beta_0\sim\operatorname{N}\Bigl(\beta_0,\ \frac{\sigma^2\sum_ix_i^2}{nS_{xx}}\Bigr),
and with S^2=\frac1{n-2}\sum(Y_i-\widehat\beta_0-\widehat\beta_1x_i)^2,
\frac{(n-2)S^2}{\sigma^2}\sim\chi^2_{n-2},
independently of (\widehat\beta_0,\widehat\beta_1). In particular S^2 is unbiased for \sigma^2. Moreover \widehat\beta_1 and \overline Y are independent.
Used in
- Taught in Linear regression, where the derivation and examples are.
- All 5 questions on Linear regression
See also
Least-squares line · Least-squares estimators are unbiased · Simple linear regression · Slope t pivot · Coefficient of determination · Prediction interval