The rates of return per dollar invested in two common stocks over a given investment period can be viewed as the outcome of a bivariate normal distribution \(N(\boldsymbol{\mu}, \boldsymbol{\Sigma})\). The rates are independent between investment periods. An investment firm intends to use a random sample from the \(N(\boldsymbol{\mu}, \boldsymbol{\Sigma})\) population distribution of rates of return to generate estimates of the expected rates of return, \(\boldsymbol{\mu}\), as well as the variances in the rates of return, given by the diagonal of \(\mathbf{\Sigma}\).

a. Find a minimal, complete (vector) sufficient statistic for \(N(\boldsymbol{\mu}, \mathbf{\Sigma})\).

b. Define the MVUE for \(\boldsymbol{\mu}\).

c. Define the MVUE for \(\mathbf{\Sigma}\) and for \(\operatorname{diag}(\mathbf{\Sigma})\).

d. Define the MVUE for the vector \(\left(\mu_{1}, \mu_{2}, \Sigma_{1}^{2}, \Sigma_{2}^{2}ight)\).

e. A random sample of size 50 has an outcome that is summarized by: \(\overline{\mathbf{x}}=[.048 .077]^{\prime}, s_{1}^{2}=.5 \times 10^{-3}, s_{2}^{2}=\) \(.3 \times 10^{-4}\), and \(s_{12}=.2 \times 10^{-4}\). Calculate the MVUE outcome for \(\left[\mu_{1}, \mu_{2}, \Sigma_{1}^{2}, \Sigma_{2}^{2}ight]^{\prime}\).

f. Is the MVUE of \(\left[\mu_{1}, \mu_{2}, \Sigma_{1}^{2}, \Sigma_{2}^{2}ight]^{\prime}\) consistent?

g. If an investor invests \(\$ 500\) in each of the two investments, what is the MVUE of her expected dollar return on the investment during the investment period under consideration?