7.4 Exercises

EXERCISE 7.1   Use Theorem 7.1 to derive a test for testing the hypothesis that a dice is balanced, based on $n$ tosses of that dice. (Hint: use the multinomial probability function.)

EXERCISE 7.2   Consider $N_{3}(\mu,\Sigma)$. Formulate the hypothesis $H_{0} : \mu_{1}=\mu_{2}=\mu_{3}$ in terms of $\data{A}\mu =a$.

EXERCISE 7.3   Simulate a normal sample with $\mu={1\choose 2}$ and $\Sigma=\left( {1\atop
0.5}{0.5\atop 2}\right)$ and test $H_{0} : 2\mu_{1}-\mu_{2}=0.2$ first with $\Sigma$ known and then with $\Sigma$ unknown. Compare the results.

EXERCISE 7.4   Derive expression (7.3) for the likelihood ratio test statistic in Test Problem 2.

EXERCISE 7.5   With the simulated data set of Example 7.14, test the hypothesis of equality of the covariance matrices.

EXERCISE 7.6   In the U.S. companies data set, test the equality of means between the energy and manufacturing sectors, taking the full vector of observations $X_1$ to $X_6$. Derive the simultaneous confidence intervals for the differences.

EXERCISE 7.7   Let $X\sim N_2(\mu,\Sigma)$ where $\Sigma$ is known to be $\Sigma =\left(\begin{array}{rr}
2 &-1\\
-1 & 2
\end{array}\right)
$. We have an i.i.d. sample of size $n=6$ providing $\bar{x}^{\top} = \left(1\ \frac{1}{2}\right)$. Solve the following test problems ($\alpha = 0.05$):

a) $H_0$: $\mu =\left(2, \frac{2}{3}\right)^{\top}$ $H_1$: $\mu \neq\left(2, \frac{2}{3}\right)^{\top}$
b) $H_0$: $\mu_1+\mu_2=\frac{7}{2}$ $H_1$: $\mu_1+\mu_2\neq\frac{7}{2}$
c) $H_0$: $\mu_1-\mu_2=\frac{1}{2}$ $H_1$: $\mu_1-\mu_2\neq\frac{1}{2}$
d) $H_0$: $\mu_1=2$ $H_1$: $\mu_1\neq 2$

For each case, represent the rejection region graphically (comment!).

EXERCISE 7.8   Repeat the preceeding exercise with $\Sigma$ unknown and $S=\left(\begin{array}{rr}
2 & -1\\
-1 & 2
\end{array}\right)$. Compare the results.

EXERCISE 7.9   Consider $X\sim N_3(\mu,\Sigma)$. An i.i.d. sample of size $n=10$ provides:

\begin{eqnarray*}
\bar{x} &=&(1, 0, 2)^{\top}\\
S&=&\left(\begin{array}{ccc}
3&2&1\\
2&3&1\\
1&1&4
\end{array}\right).
\end{eqnarray*}



a)
Knowing that the eigenvalues of $S$ are integers, describe a 95% confidence region for $\mu$. (Hint: to compute eigenvalues use $\vert S\vert=\prod\limits_{j=1}^3\lambda_j$ and $\textrm{tr}(S)=\sum\limits_{j=1}^3\lambda_j$).
b)
Calculate the simultaneous confidence intervals for $\mu_1,\mu_2$ and $\mu_3$.
c)
Can we assert that $\mu_1$ is an average of $\mu_2$ and $\mu_3$?

EXERCISE 7.10   Consider two independent i.i.d. samples, each of size 10, from two bivariate normal populations. The results are summarized below:

\begin{displaymath}\bar{x}_1=(3, 1)^{\top};\ \bar{x}_2=(1, 1)^{\top}\end{displaymath}


\begin{displaymath}
S_1=\left(\begin{array}{rr}
4 &-1\\
-1 & 2
\end{array}\righ...
..._2=\left(\begin{array}{rr}
2 &-2\\
-2 & 4
\end{array}\right).
\end{displaymath}

Provide a solution to the following tests:

a) $H_0$: $\mu_1 = \mu_2$ $H_1$: $\mu_1 \not= \mu_2$
b) $H_0$: $\mu_{11}= \mu_{21}$ $H_1$: $\mu_{11}\not= \mu_{21}$
c) $H_0$: $\mu_{12}= \mu_{22}$ $H_1$: $\mu_{12}\not= \mu_{22}$

Compare the solutions and comment.

EXERCISE 7.11   Prove expression (7.4) in the Test Problem 2 with log-likelihoods $\ell_0^*$ and $\ell_1^*$. (Hint: use (2.29).

EXERCISE 7.12   Assume that $ X \sim N_p (\mu, \Sigma) $ where $\Sigma$ is unknown.
a)
Derive the log likelihood ratio test for testing the independence of the $p$ components, that is $H_0:\ \Sigma$ is a diagonal matrix. (Solution: $-2\log \lambda =-n\log \vert R\vert$ where $R$ is the correlation matrix, which is asymptotically a $\chi^2_{\frac{1}{2}p(p-1)}$ under $H_0$).
b)
Assume that $\Sigma$ is a diagonal matrix (all the variables are independent). Can an asymptotic test for $H_0:\ \mu=\mu_o$ against $H_1:\ \mu\neq\mu_o$ be derived? How would this compare to $p$ independent univariate $t-$tests on each $\mu_j$?
c)
Show an easy derivation of an asymptotic test for testing the equality of the $p$ means (Hint: use $(C\bar{X})^{\top}(CSC^{\top})^{-1}C\bar{X} \to \chi^2_{p-1}$ where $S =\textrm{diag}
(s_{11},\ldots ,s_{pp})$ and $C$ is defined as in (7.10)). Compare this to the simple ANOVA procedure used in Section 3.5.

EXERCISE 7.13   The yields of wheat have been measured in 30 parcels that have been randomly attributed to 3 lots prepared by one of 3 different fertilizer A B and C. The data are

Fertilizer Yield A B C
1 4 6 2
2 3 7 1
3 2 7 1
4 5 5 1
5 4 5 3
6 4 5 4
7 3 8 3
8 3 9 3
9 3 9 2
10 1 6 2

Using Exercise 7.12,

a)
test the independence between the 3 variables.
b)
test whether $\mu^{\top}=[2\ 6\ 4]$ and compare this to the 3 univariate $t-$tests.
c)
test whether $\mu_1=\mu_2=\mu_3$ using simple ANOVA and the $\chi^2$ approximation.

EXERCISE 7.14   Consider an i.i.d. sample of size $n=5$ from a bivariate normal distribution

\begin{displaymath}X\sim N_2\left(\mu,
\left(\begin{array}{cc}
3 & \rho\\
\rho & 1
\end{array}\right)\right)\end{displaymath}

where $\rho$ is a known parameter. Suppose $\bar{x}^{\top} =(1\ 0)$. For what value of $\rho$ would the hypothesis $H_0:\ \mu^{\top}=(0\ 0)$ be rejected in favor of $H_1:\ \mu^{\top} \not= (0\ 0)$ (at the 5% level)?

EXERCISE 7.15   Using Example 7.14, test the last two cases described there and test the sample number one ($n_1=30$), to see if they are from a normal population with $\Sigma =4I_4$ (the sample covariance matrix to be used is given by $S_1$).

EXERCISE 7.16   Consider the bank data set. For the counterfeit bank notes, we want to know if the length of the diagonal ($X_6$) can be predicted by a linear model in $X_1$ to $X_5$. Estimate the linear model and test if the coefficients are significantly different from zero.

EXERCISE 7.17   In Example 7.10, can you predict the vocabulary score of the children in eleventh grade, by knowing the results from grades 8-9 and 10? Estimate a linear model and test its significance.

EXERCISE 7.18   Test the equality of the covariance matrices from the two groups in the WAIS subtest (Example 7.19).

EXERCISE 7.19   Prove expressions (7.21), (7.22) and (7.23).

EXERCISE 7.20   Using Theorem 6.3 and expression (7.16), construct an asymptotic rejection region of size $\alpha$ for testing, in a general model $f(x,\theta)$, with $\theta \in \mathbb{R}^k,\\
H_0: \theta=\theta_0$ against $H_1: \theta \not= \theta_0$.

EXERCISE 7.21   Exercise 6.5 considered the pdf $f(x_1,x_2)=\frac{1}{\theta_1^2\theta_2^2 x_2}e^
{-\left(\frac{x_1}{\theta_1x_2}+\frac{x_2}{\theta_1\theta_2}\right)}$
$x_1,x_2>0$. Solve the problem of testing $H_0: \theta^{\top}=\left(\theta_{01},\theta_{02}\right)$ from an iid sample of size $n$ on $x=(x_1, x_2)^{\top}$, where $n$ is large.

EXERCISE 7.22   In Olkin and Veath (1980), the evolution of citrate concentrations in plasma is observed at 3 different times of day, $X_1$ (8 am), $X_2$ (11 am) and $X_3$ (3 pm), for two groups of patients who follow a different diets. (The patients were randomly attributed to each group under a balanced design $n_1=n_2=5$).
The data are:

Group $X_1$(8 am) $X_2$(11 am) $X_3$(3 pm)
  125 137 121
  144 173 147
I 105 119 125
  151 149 128
  137 139 109
  93 121 107
  116 135 106
II 109 83 100
  89 95 83
  116 128 100

Test if the profiles of the groups are parallel, if they are at the same level and if they are horizontal.