Outline of GLMs

which becomes a one parameter exponential family in θi if the scale parameter φ is known. This is simpler than McCullagh and Nelder’s version: they use a(φ) or even ai(φ) in place of φ. The ai(φ) version is useful for data whose precision varies in a known way, as for example when Yi is an average of ni iid random variables. There are two simple examples to keep in mind. Linear regression has Yi ∼ N(θi, σ2) where θi = xiβ. Logistic regression has Yi ∼ Bin(m, θi) where θi = (1 + exp(−xiβ)). It is a good exercise to write these models out as in (1), and identify φ, b and c. Consider also what happens for the models Yi ∼ N(θi, σ2 i ) and Yi ∼ Bin(mi, θi). The mean and variance of Yi can be determined from its distribution given in equation (1). They must therefore be expressable in terms of φ, θi, and the functions b(·) and c(·, ·). In fact,