Random and Mixed Effects Models

In Example 5.1:

Let $A=\left(a_{i j}\right)$ be a nonsingular $n \times n$ matrix with $a_{i i}=a$ and $a_{i j}=b$ for all $i \neq j$. Determine the elements of $A^{-1}$.

Verify the UMVU estimator of $\sigma_{A}^{2} / \sigma^{2}$ given in Example 5.1.

Obtain the joint density of the $X_{i j}$ in Example 5.1 in the unbalanced case in which $j=1, \ldots , n_{i}$, with the $n_{i}$ not all equal, and determine a minimal set of sufficient statistics (which depends on the number of distinct values of $n_{i}$ ).

In the balanced one-way layout of Example 5.1, determine $\lim \mathbb {P}\left(\hat{\sigma }_{A}^{2}<0\right)$ as $n \rightarrow \infty$ for $\sigma_{A}^{2} / \sigma^{2}=0,0.2,0.5,1$, and $s=3,4,5,6$.

In the preceding problem, calculate values of $\mathbb {P}\left(\hat{\sigma }_{A}^{2}<0\right)$ for finite $n$. When would you expect negative estimates to be a problem?

The following problem shows that in Examples 5.1-5.3 every unbiased estimator of the variance components (except $\sigma^{2}$ ) takes on negative values. (For some related results, see Pukelsheim 1981.) Let $X$ have distribution $P \in \mathcal{P}$ and suppose that $T$ is a complete sufficient statistic for $\mathcal{P}$. If $g(P)$ is any $U$-estimable function defined over $\mathcal{P}$ and its UMVU estimator $\eta (T)$ takes on negative values with probability $>0$, then show that this is true of every unbiased estimator of $g(P)$.

Modify the car illustration of Example 5.1 so that it illustrates (5.5).

In Example 5.2, define a linear transformation of the $X_{i j k}$ leading to the joint distribution of the $Z_{i j k}$ stated in connection with (5.6), and verify the complete sufficient statistics (5.7).

In Example 5.2, obtain the UMVU estimators of the variance components $\sigma_{A}^{2}, \sigma_{B}^{2}$, and $\sigma^{2}$ when $\sigma_{C}^{2}=0$, and compare them to those obtained without this assumption.

For the $X_{i j k}$ given in (5.8), determine a transformation taking them to variables $Z_{i j k}$ with the distribution stated in Example 5.3.

In Example 5.3, obtain the UMVU estimators of the variance components $\sigma_{A}^{2}, \sigma_{B}^{2}$, and $\sigma^{2}$.

In Example 5.3, obtain the UMVU estimators of $\sigma_{A}^{2}$ and $\sigma^{2}$ when $\sigma_{B}^{2}=0$ so that the $B$ terms in (5.8) drop out, and compare them with those of Problem 5.12.

In Example 5.4:

A general class of models containing linear models of Types I and II, and mixed models as special cases assumes that the $1 \times n$ observation vector $\mathbf{X}$ is normally distributed with mean $\boldsymbol {\theta } A$ as in (4.13) and with covariance matrix $\Sigma_{i=1}^{m} \gamma_{i} V_{i}$ where the $\gamma$ 's are the components of variance and the $V_{i}$ 's are known symmetric positive semidefinite $n \times n$ matrices. Show that the following models are of this type and in each case specify the $\gamma$ 's and $V$ 's:

Consider a nested three-way layout with

$(i=1, \ldots , I ; j=1, \ldots , J ; k=1, \ldots , K ; l=1, \ldots , n)$ in the versions

For the situation of Example 5.5, relax the assumption of normality to only assume that $A_{i}$ and $U_{i j}$ have zero means and finite second moments. Show that among all linear estimators (of the form $\sum c_{i j} x_{i j}, c_{i j}$ known), the UMVU estimator of $\mu +\alpha_{i}$ (the best linear predictor) is given by (5.14).