Question

how to show they are equivalent?

3. (No R Required) Cook's distance has the equivalent formulae (8 – B«)'(x+x)(– B6) Di = (1) pMSres = r? hii p 1 - hii (2) where r; denotes studentized residuals. Show that (1) and (2) are equivalent. You may use the following without proof: Ê - B(i) = (1 - hii)-4(X'X)-X;li. (3)

Answer 1

$D_i=\frac{\left(\hat{\boldsymbol{\beta}}-\hat{\boldsymbol{\beta}}_{(i)} \right )^\prime ({\bf X}^\prime {\bf X})\left(\hat{\boldsymbol{\beta}}-\hat{\boldsymbol{\beta}}_{(i)} \right )}{pMS_{res}}=\frac{(1-h_{ii})^{-2}e_i^2{\bf X}_i^\prime({\bf X}^\prime{\bf X}){^{-1}\bf X}_i}{pMS_{res}}\\\\ =\frac{(1-h_{ii})^{-2}e_i^2h_{ii}}{pMS_{res}}=\frac{1}{p}\times\frac{e_i^2}{MS_{res}(1-h_{ii})}\times \frac{h_{ii}}{1-h_{ii}}=\frac{r_i^2}{p}\times \frac{h_{ii}}{1-h_{ii}};\\\\ ~where,~h_{ii}={\bf X}_i^\prime({\bf X}^\prime{\bf X})^{-1}{\bf X}_i=(i,i)th~diagonal~element~{\bf X}^\prime\\\\ r_i=studentized~residual=\frac{e_i}{\sqrt{MS_{res}(1-h_{ii})}}$