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 Incoming links: Gamma regression with logarithmic link

Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 1 / 13J / iv / Solution Created 2026-09-24 Updated 2026-09-29

A suitable generalized linear model is a Gamma regression with logarithmic link:

Y_{i} ∣ x_{i} has a gamma distribution, E (Y_{i} ∣ x_{i}) = μ_{i}, Var (Y_{i} ∣ x_{i}) = ϕ μ_{i}^{2},

with

lo g μ_{i} = β_{0} + β_{K} numeric (Kilometres_{i}) + β_{Brand_{i}} + β_{Bonus_{i}} .

The gamma distribution is appropriate for a positive, continuous, right-skewed payment, and its variance function allows the conditional variance to grow like the square of the conditional mean instead of assuming constant variance. The logarithmic link function also guarantees positive fitted means and represents covariate effects multiplicatively, which accords with the Box–Cox transformation evidence for a logarithmic scale.

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Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 218 / 3 / a / Solution Created 2026-09-28 Updated 2026-10-03

Let

T_{i} > 0

be record time and let

x_{i} = (1, c_{i}, d_{i})^{T}

contain standardized climb and distance. Model 1 is the normal linear model

T_{i} = x_{i}^{T} β + ε_{i}, ε_{i} \sim iid N (0, σ^{2}) .

Model 2 applies the same model after a logarithmic transformation:

lo g T_{i} = x_{i}^{T} α + e_{i}, e_{i} \sim iid N (0, τ^{2}),

so

T_{i}

is conditionally log-normal. Model 3 is a Gamma regression with logarithmic link:

E (T_{i} ∣ x_{i}) = μ_{i}, Var (T_{i} ∣ x_{i}) = ϕ μ_{i}^{2}, lo g μ_{i} = x_{i}^{T} γ .

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