STAT 350

Postscript version of these questions

Assignment 1

1.

Suppose $Z_1,\ldots,Z_{10}$ are independent random variables each having a N(0,6) distribution. Let ${\bar Z} = \sum Z_i/10$, $U=\sum_{i=1}^4 Z_i^2/6$, $V=\sum_{i=5}^{10}Z_i^2/6$, $X=Z_1/\sqrt{V}$ and Y=(3U)/(2V). Give the names for the distributions of each of ${\bar Z}$, U, V, X and Y and use tables to find $P(\vert{\bar Z}\vert > 1)$, $P(U \le 9.49)$, $P(-1.2 \le X \le 1.2)$, P( Y > 6.23), $P(U \le V)$, $P(Z_1 - 2 Z_2 \ge 10)$.

2.

A new process for measuring the concentration of a chemical in water is being investigated. A total of n samples are prepared in which the concentrations are the known numbers x_i for $i=1,\ldots,n$; the new process is used to measure the concentrations for these samples. It is thought likely that the concentrations measured by the new process, which we denote Y_i, will be related to the true concentrations via

$$Y_i = \beta x_i + \epsilon_i$$

where the $\epsilon_i$ are independent, have mean 0 and all have the same variance $\sigma^2$ which is unknown.

(a)

If this model is fitted by least squares, (that is by minimizing $\sum_i(Y_i-\beta x_i)^2$) show that the least squares estimate of $\beta$ is

$$\hat\beta = \frac{\sum x_iY_i}{\sum x_i^2} \, .$$

(b)

Show that the estimator in part (a) is unbiased.

(c)

Compute (give a formula for) the standard error of $\hat\beta$.

(d)

The error sum of squares for this model is $\sum(Y_i -\hat\beta x_i)^2$ which may be shown to have n-1 degrees of freedom. If the x_i are the numbers 1, 2, 3 and 4, $\hat\beta = 1$ and the error sum of squares is 0.12 find a 95% confidence interval for $\beta$ and explain what further assumptions you must make to do so.

(e)

Show that the estimator

$$\tilde\beta = \frac{\sum Y_i}{\sum x_i}$$

is also unbiased.

(f)

Compute (give a formula for) the standard error of $\tilde\beta$. Which is bigger, the standard error of $\hat\beta$ or that of $\tilde\beta$?

(g)

Show that the mle of $\beta$ in this model is $\hat\beta$, the least squares estimate, if the $\epsilon_i$ have normal distributions.

3.

Consider the two-way layout without replicates. We have data Y_ij for $i=1,\ldots I$ and $j=1,\ldots,J$. We generally fit a so-called additive model

$$Y_{ij} = \mu + \rho_i+\gamma_j + \epsilon_{ij}$$

In the following questions consider the case I=2 and J=3.

(a)

If we treat $\mu$, $\rho_1$, $\rho_2$, $\gamma_1$, $\gamma_2$ and $\gamma_3$ as the entries in the parameter vector $\beta$ what is the design matrix X=X_a and what is the rank of X_a?

(b)

What is the determinant of the matrix X_a^T X_a? Is this matrix invertible? How many solutions do the normal equations have?

(c)

Usually we impose the restrictions $\rho_1+\rho_2=0$ and $\gamma_1+\gamma_2+\gamma_3=0$. Use these restrictions to eliminate $\rho_2$ and $\gamma_3$ from the model equation and, for the parameter vector $\beta^T = (\mu,\rho_1,\gamma_1,\gamma_2)$ find the design matrix X_b.

(d)

An alternate set of restrictions is called corner point coding where we assume $\rho_1=\gamma_1=0$. With this restriction and the parameter vector $\beta^T = (\mu,\rho_2,\gamma_2,\gamma_3)$ what is the design matrix X_c?

(e)

Show that the three design matrices have the same column space by finding a matrix A such that X_a = X_b A and similarly for X_b and X_c and for X_a and X_c.

(f)

Use the previous part to show that the vectors of fitted values $\hat Y$ will be the same for any solution of the normal equations for any of the three design matrices.

4.

From the text question 1.19, 1.23, 2.13 a and b and 2.23 a, b and c. In 2.23 c give a P-value and interpret this P-value.

DUE: Friday, 15 January.