Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 41 / 1 / d / i / Solution

Write ${\mu }_i=\alpha +\beta z_i+{\gamma }^{\mathsf{T}}{x}_i$ and let $f_i(y)={\sigma }^{-1}\varphi ((y-{\mu }_i)/\sigma )$ be the normal distribution density. Set $q_j(y)=\mathrm{expit}({\eta }_j+\theta y)$, where $q_j$ is the probability that attempt $j$ is unsuccessful conditional on the preceding attempts being unsuccessful. This is a sequential response selection model. For first-attempt success, the observed outcome and attempt count contributeThis likelihood conditions on baseline and assignment; the displayed response probabilities do not use $d_i$. No factor for random allocation is needed when conditioning on its realized value.