Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 221 / 4 / c / ii / Solution

Fix $\rho$. For an observation with covariates $x$, putand let $\Phi$ be the standard-normal distribution function. The four conditional cell probabilities arewhere the first index is $A$ and the second is $Y$.

Define $({\hat{\alpha }}_{\rho },{\hat{\beta }}_{\rho },{\hat{\gamma }}_{\rho })$ as any maximizer of the log likelihoodThen ${\hat{\beta }}_{\rho }$ is the requested estimator for the fixed sensitivity value $\rho$.