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IAI Actuarial Core Principles · Actuarial Statistics · Generalised linear models

An insurer models the number of motor claims per policy using a Poisson GLM with a log link. The exposure of each policy is the fraction of a year it was in force. What is the standard way to allow for differing exposures?

Include log(exposure) as an offset with coefficient fixed at 1. Under a log link the expected count is exposure times the exponential of the linear predictor, so the log of exposure enters with a fixed unit coefficient, keeping the model for counts proportional to time at risk.

  1. ADivide the claim count by exposure and fit a Poisson model to the resulting rate
  2. BInclude log(exposure) as an offset with coefficient fixed at 1Correct
  3. CInclude exposure as an ordinary covariate with an estimated coefficient
  4. DWeight each observation by the log of exposure
  5. Use an identity link so that exposure enters additively

Explanation

With a log link, E[N] = exposure x exp(linear predictor), so log(exposure) enters the linear predictor with its coefficient fixed at 1, which is an offset. Fitting a Poisson model to the rate breaks the Poisson count distribution assumption, and estimating the coefficient freely does not impose proportionality.

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