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IAI Actuarial Core Principles · Actuarial Statistics · Generalised linear models

A claim count N follows a Poisson distribution with mean μ. Written in exponential family form, what are the canonical parameter θ, the function b(θ) and the canonical link?

The Poisson has θ = ln μ and b(θ) = e^θ, since the pmf is exp{y ln μ − μ − ln y!}. The canonical link sets g(μ) = θ, which gives the log link. This also gives variance function V(μ) = μ.

  1. Aθ = μ, b(θ) = θ², link g(μ) = μ
  2. Bθ = ln μ, b(θ) = e^θ, link g(μ) = ln μCorrect
  3. Cθ = ln μ, b(θ) = ln θ, link g(μ) = 1/μ
  4. Dθ = e^μ, b(θ) = ln θ, link g(μ) = ln μ
  5. θ = 1/μ, b(θ) = e^θ, link g(μ) = 1/μ

Explanation

The Poisson pmf is exp{ y ln μ − μ − ln y! }. So θ = ln μ and b(θ) = μ = e^θ. The canonical link is the one with g(μ) = θ, which is the log link. Then b'(θ) = e^θ = μ and b''(θ) = μ, as expected.

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