Suppose $Y = X \beta + \epsilon$ where $\epsilon \sim N(0, \sigma^2I)$. Show that for the internally studentised residuals $r_i$ defined as $$ r_i = \frac{\hat{\epsilon}_i}{\hat{\sigma}\sqrt{1-h_{ii}}} $$ the expression $\frac{r_i^2}{(n-p)}$ has the $Beta(\frac{1}{2},\frac{(n-p-1)}{2})$ distribution where $p = rank(X)$.

How can I prove it?

I have the following hints:

(i) Recall that the $Beta(p,q)$ distribution is defined as the distribution of $\frac{U}{U+V}$ where $U$ and $V$ are independent, and $U \sim Gamma(p,\lambda), V \sim Gamma(q,\lambda)$.

(ii) Let $e_i$ be the $n$-dimensional vector whose only nonzero component is a $1$ in the $i$-th position, and note that $\hat{\epsilon}_i = e_i^T (I-H)Y$.

(iii) Define $P = I - H$ and $P_1 = \frac{Pe_i e_i^TP}{1-h_{ii}}$ and $U = Y'P_1Y$ and $V = Y'(P-P_1)Y$. Show that $U$ and $V$ are independent and have Gamma distributions.


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.