Skip to content

IAI Actuarial Core Principles · Actuarial Statistics · Generalised linear models

For a Bernoulli response with success probability p, the pmf is written in exponential family form. Which choice gives the canonical parameter and the canonical link function?

The canonical parameter is θ = ln(p/(1−p)), so the canonical link is the logit. Writing the Bernoulli pmf as exp{y ln(p/(1−p)) + ln(1−p)} shows this directly. Probit and complementary log-log links are usable but not canonical.

  1. Aθ = ln(p/(1 − p)), the logit linkCorrect
  2. Bθ = ln p, the log link
  3. Cθ = p, the identity link
  4. Dθ = Φ⁻¹(p), the probit link
  5. θ = ln(−ln(1 − p)), the complementary log-log link

Explanation

The pmf is p^y(1−p)^(1−y) = exp{ y ln(p/(1−p)) + ln(1−p) }. Hence θ = ln(p/(1−p)) and b(θ) = ln(1+e^θ). The canonical link equals θ as a function of the mean p, which is the logit. Probit and complementary log-log are valid links but not canonical.

Did you get it right without looking?

One question tells you little. A timed set on Generalised linear models shows your real accuracy, how long you take and where you lose marks.

Practise Generalised linear modelsFree diagnostic mock

More Generalised linear models questions