Table 1 Distributional assumptions regarding the underlying age-specific malaria FOI
Distribution | \(\varvec{\theta }\) | \(h(a_{ij}; \varvec{\theta })\) | \(\lambda _0(a_{ij})\) |
|---|---|---|---|
Exponential | \(\theta _1 > 0\) | \(\log (\theta _1 a_{ij})\) | \(\theta _1\) |
Weibull | \(\theta _1, \theta _2 > 0\) | \(\log (\theta _1 a_{ij}^{\theta _2})\) | \(\theta _1 \theta _2 a_{ij}^{\theta _2-1}\) |
Gompertz | \(\theta _1 > 0, -\infty< \theta _2 < +\infty\) | \(\log \left[ \frac{\theta _1}{\theta _2} \left( e^{\theta _2 a_{ij}}-1\right) \right]\) | \(\theta _1 e^{\theta _2 a_{ij}}\) |
Log-logistic | \(\theta _1, \theta _2 > 0\) | \(\log \left\{ \log \left[ 1+\left( \theta _1 a_{ij}\right) ^{\theta _2}\right] \right\}\) | \(\frac{\theta _1 \theta _2 (\theta _1 a_{ij})^{\theta _2 - 1}}{1 + (\theta _1 a_{ij})^{\theta _2}}\) |
Fractional polynomial | \(\theta _2 < 0\) | \(\theta _2 a_{ij}^{-1}\) | \(-\theta _2 a_{ij}^{-2} e^{\theta _2 a_{ij}^{-1}}\) |