STAT 350

Postscript version of these questions

Assignment 5

Part A

1.

In this question you will derive some of the formulas for case deleted statistics. Suppose that X is an $n\times p$ design matrix. Let x_i^T be the $i^{\rm th}$ row of X (so that, x_i is a column vector or dimension p). Let X_(i) be the design matrix with case i deleted and Y_(i) the vector of n-1 responses with case i deleted.

(a)

Use partitioned matrices to show

$$X_{(i)}^T Y_{(i)} = X^TY -x_iY_i \, .$$

(b)

Show

$$X_{(i)}^TX_{(i)} = X^TX - x_ix_i^T \, .$$

(c)

Suppose B is an invertible symmetric $p\times p$ matrix and v is a column vector of dimension p. Show, by direct multiplication, that (B-vv^T)^-1 is of the form

B^-1 + r B^-1vv^T B^-1

and give a formula for the scalar r.

(d)

Apply to previous part to show

(X_(i)^TX_(i))^-1 = (X^TX)^-1 + r_i(X^TX)^-1 x_i x_i^t (X^TX)^-1

and give a formula for r_i in terms of the leverage h_ii = x_i^T (X^TX)^-1 x_i.

(e)

Show 1+h_iir_i = r_i.

(f)

Show that

$$\begin{eqnarray*}\hat\beta_{(i)} & = & \hat\beta +r_i (X^TX)^{-1} x_i x_i^T \hat... ...i \\ & = & \hat\beta - r_i (X^TX)^{-1} x_i \hat\epsilon_i \, . \end{eqnarray*}$$

(g)

Deduce that $\hat\mu_{(i)} = \hat\mu_i - r_i h_{ii} \hat\epsilon_i$.

(h)

Show that the $i^{\mbox{th}}$ PRESS residual $Y_i - \hat\mu_{(i)}$ is given by

$$Y_i - \hat\mu_{(i)} = r_i\hat\epsilon_i \,$$

(i)

Derive the formula for the $i^{\mbox{th}}$ externally studentized (case deleted) residual.

2.

For the Nitrogen output in Wallabies data set from Assignment 3 do forward, backward, stepwise and all subsets regression.

3.

Problem 23.12, 23.13 parts c,d,e,f,g only and 23.14.

4.

Problem 25.11 parts a, b anc c, 25.12 part c.