Let $\{X_i\}_{i=1}^n\overset{iid}{\sim}\mathcal{N}(\mu,\sigma^2)$. Let $\{b_i\}_{i=1}^n$ be a sequence of numbers so that $\sum_{i=1}^nb_i=0$ and $\sum_{i=1}^nb^2_i=1$. Define $$S^2=\sum_{i=1}^n(X_i-\bar{X})^2 \text{ and } Y=\sum_{i=1}^nb_iX_i$$.

What can you say about the joint distribution of $Y$ and $S^2-Y^2$?

I only can get the marginal distribution of $Y\sim\mathcal{N}(0,\sigma^2)$, $S^2\sim\sigma^2\chi_{n-1}^2$ and $Y^2/\sigma^2\sim\chi^2_{1}$. What can we say about the joint distribution of $Y$ and $S^2-Y^2$? I tried Basu's theorem, based on the fact that $S^2$ is complete and sufficient for $\sigma^2$ but I cannot show anything using this...

  • 1
    $\begingroup$ Hint: you can simultaneously diagonalize the quadratic forms $S^2$ and $Y^2.$ $\endgroup$
    – whuber
    Feb 16 '21 at 16:38
  • $\begingroup$ So $Y^2=b^TXX^Tb$ and $S^2 = X^TX-2X^T\bar{X}+\bar{X}^T\bar{X}$? $\endgroup$
    – Tan
    Feb 17 '21 at 0:39
  • $\begingroup$ Write the forms as matrices. For instance, $Y^2 = X^\prime(bb^\prime)X$ has matrix $bb^\prime.$ $\endgroup$
    – whuber
    Feb 17 '21 at 1:00

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.