Exercises for Wed Mar 13

1. On Wed Mar 6 we discussed the first part of Ch 5, with tolerance intervals around parametric models, along with examples. I also went through parts of extra exercises, skiing days at Bj?rnholt plus exam 2015, exercise 3, the Danish melanoma study.

2. I've uploaded *com19b* (skiing days at Bj?rnholt, two models) and *com20a* (for the Danish melanoma data). Run them, read and modify, part by part, and do more.

3. For Wed Mar 13, do these extra exercises, related to tolerance regions around models.

(a) "Grow the normal" by including the extra parameter \gamma, in F(y,\xi,\sigma,\gamma) = \Phi( (y-\xi)/\sigma )^\gamma. Draw densities for \gamma equal to 0.9, 1.0, 1.1. Compute skewness as a function of \gamma. How much must \gamma differ from \gamma_0 = 1, in order for the three-parameter model to be better than then classic school-book normal model?

(b) Go skiing to Bj?rnholt. Take the home model to be classic linear regression, y_i = a + b x_i + \sigma \eps_i, with the \eps_i iid and standard normal. Then "grow the model" in at least two directions, and for each do the Fisher information matrix calculations, leading to J_\wide and tolerance levels: (i) when including a quadratic term, as in a + b x_i + c x_i^2; (ii) when including a heteroscedastic term in the variance, with \sigma_i = \sigma_0\exp(\gamma w_i), with w_i = (x_i - \bar x)/s_x.