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>
$$
T_i=x_i^T\beta+\varepsilon_i,
\qquad \varepsilon_i\overset{\mathrm{iid}}\sim N(0,\sigma^2).
$$
Model 2 applies the same model after a <logarithmic transformation>:
$$
\log T_i=x_i^T\alpha+e_i,
\qquad e_i\overset{\mathrm{iid}}\sim N(0,\tau^2),
$$
so $T_i$ is conditionally log-normal. Model 3 is a <Gamma regression with logarithmic link>:
$$
\mathbb E(T_i\mid x_i)=\mu_i,
\qquad
\operatorname{Var}(T_i\mid x_i)=\phi\mu_i^2,
\qquad
\log\mu_i=x_i^T\gamma.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 218 / 3 / a / Solution

{c}

A Gamma regression with logarithmic link models a positive response with $\operatorname{Var}(Y_i\mid X_i)=\phi\mu_i^2$ and $\log\mu_i=x_i^T\beta$.


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 Incoming links: Gamma regression with logarithmic link