Exercises 1

Exercise 1.1

Linear mixed effect regression model specifies that response vectors for individual \(i\), \(\boldsymbol{Y}_i \in \mathbb{R}^k\), are Gaussian. The model includes model matrix \(\mathbf{X}_i\) with fixed effect coefficients \(\boldsymbol{\beta}\), and another \(k\times l\) model matrix \(\mathbf{Z}_i\) with random effects. The hierarchical formulation of the model is \[\begin{align*} \boldsymbol{Y}_i \mid \mathcal{B}_i=\boldsymbol{b}_i &\sim \mathsf{Gauss}_k(\mathbf{X}_i\boldsymbol{\beta} + \mathbf{Z}_i\boldsymbol{b}_i, \sigma^2 \mathbf{I}_k) \\ \mathcal{B}_i & \sim \mathsf{Gauss}_l(\boldsymbol{0}_k, \boldsymbol{\Omega}) \end{align*}\]

  1. Using the tower property, derive the marginal mean and covariance matrix of \(\boldsymbol{Y}_i\)
  2. Hence obtain the parameters of the joint distribution of \((\boldsymbol{Y}_i^\top, \mathcal{B}_i^\top)^\top\).

Exercise 1.3

Consider a simple random sample of size \(n\) from the Wald distribution, with density \[\begin{align*} f(y; \nu, \lambda) = \left(\frac{\lambda}{2\pi y^{3}}\right)^{1/2} \exp\left\{ - \frac{\lambda (y-\nu)^2}{2\nu^2y}\right\}, \qquad y > 0. \end{align*}\] for location \(\nu >0\) and shape \(\tau>0\). You may take for given that the expected value of the Wald distribution is \(\mathsf{E}(Y) = \nu\).

  1. Write down the likelihood and show that it can be written in terms of the sufficient statistics \(\sum_{i=1}^n y_i\) and \(\sum_{i=1} y_i^{-1}\).
  2. Derive the Fisher information matrix