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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.

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