832 HW8 - Due Wednesday 2020/10/21

  1. fana.pdf exercises 7 and 8.

  2. fana.pdf exercise 11.

  3. fana.pdf exercise 12.

  4. glm.pdf exercise 4. Comment on which prior distributions make computations the easiest.

  5. Fit a multivariate regression model to the four cognitive scores in the WAIS dataset to the matrix \(\mathbf X\) with \(i\)th row \(\mathbf x_{i} =\) (1,senile\(_i\)).

    1. Using a \(T^2\) test, evaluate the hypothesis that the 2nd row \(\mathbf b_2\) of \(\mathbf B\) is zero, that is, there is mean difference by senility.
    2. Obtain an unbiased estimate of the variance matrix under this model, and compare it to the unbiased estimate assuming that there is no effect of senility. Explain the differences between the two estimates.
    3. Recall that a one-dimensional factor model was a pretty good fit to these data when there was no predictor variable. Now fit the model \(\mathbf Y \sim N_{n\times p} ( \mathbf X \mathbf B , (\mathbf a\mathbf a^\top + \mathbf D^2)\otimes \mathbf I_n)\). Devise, explain and implement a procedure for fitting this model. Compare the estimated covariance model to that from the model \(\mathbf Y \sim N_{n\times p} ( \mathbf 1 \boldsymbol \mu^\top , (\mathbf a\mathbf a^\top + \mathbf D^2)\otimes \mathbf I_n)\).