832 Homework 4

1. Let \(E=CY\) where \(E[Y] = 1\mu^\top\) and \(V[Y] = \Sigma \otimes I\). Calculate the mean and variance of \(E\), and from this, the mean and variance of a row \(e_i\) of \(E\).

2. Show the eigenvalues of an orthogonal projection matrix are all either zero or one. Hence, show that the centering matrix can be written as \(C= GG^\top\) for \(G\in \mathbb R^{n\times n-1}\) such that \(G^\top G = I_{n-1}\).

3. Consider the matrix normal model \(Y \sim N_{n\times p}(1\mu^\top,\Sigma\otimes I)\).

   1. Show that \(\bar y\) and \(S=Y^\top C Y\) are complete sufficient statistics.

   2. Show that \(\bar y\) is the UMVUE for \(\mu\).

4. Show that if \(G\) and \(H\) are square roots of a \(p\times p\) non-singular covariance matrix \(\Sigma\), so that \(GG^\top = HH^\top = \Sigma\), then \(H = G O\) for some orthogonal matrix \(O\) such that \(O^\top O = I_p\).

5. Consider a random effects model for \(p\)-variate random vectors \(y_1,\ldots,y_n\) of the form \[\begin{align} y_{i} & = a_i 1_p + b + H z_i \\ a_1,\ldots, a_n & \sim \text{i.i.d.} \ N(0,\sigma^2_a) \\ z_1\ldots, z_n &\sim \text{i.i.d.} \ N_p(0, I_p). \end{align}\] where the \(a_i\)’s and \(z_i\)’s are independent, and \(b\), \(H\) and \(\sigma^2_a\) are unknown parameters.

   1. Write out a stochastic representation of the resulting \(n\times p\) data matrix \(Y\), in matrix form.
      
   2. Identify the matrix normal distribution of \(Y\) (i.e., find the mean and variance of \(Y\) given \(b\), \(H\) and \(\sigma^2_a\)).
      
   3. Find an unbiased estimate of \(b\).
      
   4. Explain why \(H\) is not identifiable. Can you estimate \(\sigma^2_a\) and \(HH^\top\)?