Fix $\rho$. For an observation with covariates $x$, put
$$
u=-x^T\alpha,
\qquad
v_a=-a\beta-x^T\gamma,
$$
and let $\Phi$ be the standard-normal distribution function. The four conditional cell probabilities are
$$
\begin{aligned}
p_{00}(x)&=F_\rho(u,v_0),\\
p_{01}(x)&=\Phi(u)-F_\rho(u,v_0),\\
p_{10}(x)&=\Phi(v_1)-F_\rho(u,v_1),\\
p_{11}(x)&=1-\Phi(u)-\Phi(v_1)+F_\rho(u,v_1),
\end{aligned}
$$
where the first index is $A$ and the second is $Y$.

Define $(\widehat\alpha_\rho,\widehat\beta_\rho,\widehat\gamma_\rho)$ as any maximizer of the <maximum likelihood estimation>[log likelihood]
$$
\sum_{i=1}^n\sum_{a,y\in\{0,1\}}
\mathbf1_{\{A_i=a,Y_i=y\}}\log p_{ay}(X_i).
$$
Then $\widehat\beta_\rho$ is the requested estimator for the fixed sensitivity value $\rho$.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-221/4/c/ii/solution