Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 218 / 1 / a

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- Solution a

Solution

Put

ϕ = 1/ γ

and

θ = - 1/ α

. The density can be written

f (y; θ, ϕ) = exp [\frac{y θ - b ( θ )}{ϕ} + c (y, ϕ)], b (θ) = - lo g (- θ),

where

c (y, ϕ) = - lo g Γ (1/ ϕ) - ϕ^{- 1} lo g ϕ + (ϕ^{- 1} - 1) lo g y .

Thus this is an exponential dispersion family. Its derivative identities give

E Y = b^{'} (θ) = - \frac{1}{θ} = α, Var (Y) = ϕ b^{''} (θ) = ϕ α^{2} = \frac{α ^{2}}{γ} .

The variance function is

V (μ) = μ^{2}

. In the usual Gamma-GLM convention the canonical inverse link is

g (μ) = \frac{1}{μ};

it differs by a minus sign from the natural parameter

θ = - 1/ μ

.

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