STAT 350

Postscript version of these questions

Assignment 2

1.

In this problem you will prove that

$$\phi(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2} \qquad -\infty < x < \infty$$

is a density.

(a)

Let $I = \int_{-\infty}^\infty \phi(x) dx$. Show that

$$I^2 = \int_{-\infty}^\infty \int_{-\infty}^\infty \phi(x)\phi(y)\, dx \, dy .$$

HINT: What is $\int_{-\infty}^\infty \phi(y)dy$ in terms of I.

(b)

Now if

$$J = \int_{-\infty}^\infty \int_{-\infty}^\infty \frac{1}{\sqrt{2\pi}} e^{-x^2/2} \frac{1}{\sqrt{2\pi}} e^{-y^2/2} \, dx \, dy$$

do the double integral J in polar co-ordinates ( $x=r\cos\theta$, $y=r\sin\theta$) to show J=1.

(c)

Deduce that $\phi$ is a density.

2.

Suppose X₁,X₂,X₃ are independent $N(\mu,\sigma^2)$ random variables, so that $X_i=\mu+\sigma Z_i$ with Z₁,Z₂,Z₃ independent standard normals.

(a)

If X^T = (X₁,X₂,X₃) and Z^T=(Z₁,Z₂,Z₃) express X in the form AZ+b for a suitable matrix A and vector b.

(b)

Show that X is $MVN_3(\mu_X,\Sigma_X)$ and identify $\mu_X$ and $\Sigma_X$.

(c)

Let $Y_i = X_i-\bar{X}$ for i=1,2,3 and $Y_4=\bar{X}$. Show that $Y\sim MVN_4(\mu_Y,\Sigma_Y)$ and find $\mu_Y$ and $\Sigma_Y$.

3.

Working with partitioned matrices. Suppose that the design matrix X is partitioned as $X=[{\bf 1}\vert X_1\vert X_2]$ where X_i has p_icolumns.

(a)

Write X^TX as a partitioned (3 rows, 3 columns) matrix.

(b)

A matrix

$$A = \left[\begin{array}{ccc} A_1 & 0 & 0 \\ 0 & A_2 & 0 \\ 0 & 0 & A_3 \end{array}\right]$$

is called block diagonal. Show that A^-1 exists if and only if each A_i^-1 exists and that then A^-1is block diagonal.

(c)

Suppose that ${\bf 1}^T X_i = 0$ for i=1,2 and X₁^TX₂=0. Show that X^TX is block diagonal and give a formula for (X^TX)^-1.

(d)

Suppose $\beta^T = [ \beta_0\vert \beta_1^T\vert\beta_2^T]$ is partitioned to conform with the partitioning of X (that is $\beta_0$ is a scalar and $\beta_i$ is a column vector of length p_i for i=1,2. Let $\tilde\beta_0$ be obtained by fitting

$$Y={\bf 1}\beta_0+\epsilon$$

by least squares, $\tilde\beta_1$ be obtained by fitting

$$Y=X_1\beta_1+\epsilon$$

and similarly for $\tilde\beta_2$. Let $\hat\beta$ be the usual least squares estimate for

$$Y=X\beta+\epsilon \, .$$

Show that $\hat\beta^T = [ \tilde\beta_0\vert \tilde\beta_1^T\vert\tilde\beta_2^T]$.

(e)

Let $\hat\mu_i$ be the vectors of fitted values corresponding to the estimates $\tilde\beta_i$ for i=1,2,3. Show that for $i\ne j$ we have $\hat\mu_i \perp \hat\mu_j$.

(f)

For the design matrix X_b of the first assignment identify X₁ and X₂ and verify the orthogonality condition of this problem.

4.

Page 321. Problem 7.33 parts a, b, e and f, 7.34 and 7.35 part a.

DUE: Wednesday, 3 February