Suppose that $ (X_1 ,Y_1...X_n,Y_n) $ is an i.i.d. random sample from a simple homoschedastic linear model $Y=\alpha +\beta X+e $ , with $e|X \sim N(0,\sigma_e^2)$.

I want to understand if $ \frac{rss}{\sigma_Y^2}=\frac{1}{\sigma_Y^2}\sum_i (Y_i - \hat{Y_i} )^2$ is independent from $\frac{nS^2}{\sigma_X^2} =\sum_i (X_i - \bar X_i )^2$.

I can easily show that $rss$ $\sim \sigma_Y^2 \chi^2_{n-2}$ and $nS^2 \sim \sigma_X^2 \chi^2_{n-1}$ via Cochran theorem.

I think that a feasible option should be Basu Theorem: both sample statistics should be ancillary, but how can I demonstrate completeness and sufficiency of one of the two?

Are any other options possible?


Your Answer

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

Browse other questions tagged or ask your own question.