Section 4.7.2 mentioned that using a gamma GLM with log-link function gives similar results to applying a normal linear model to log(y).

a. Use the delta method to show that when y has standard deviation ???? proportional to ???? (as does the gamma GLM), log(y) has approximately constant variance for small ????.

b. The gamma GLM with log link refers to log[E(yi)], whereas the ordinary linear model for the transformed response refers to E[log(yi)]. Show that if log(yi) ∼ N(????i, ????2), then log[E(yi)] = E[log(yi)] + ????2∕2.

c. For the lognormal fitted mean Li for the linear model for log(yi), explain why exp(Li) is the fitted median for the conditional distribution of yi.

Explain why the fitted median would often be more relevant than the fitted mean of that distribution.