Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 5L (Statistical Modelling)

Conditionally on $x_i$,Thus this is a probit regression with linear predictorFit $(\alpha ,\gamma )$ by maximum likelihood for Bernoulli observations, then recover

Write ${\tilde{x}}_{\ast }=(1,x_{\ast }^T{)}^T$, $\hat{\vartheta }=(\hat{\alpha },{\hat{\gamma }}^T{)}^T$, and let $\hat{V}$ estimate the asymptotic covariance of $\hat{\vartheta }$. ThenAn asymptotic confidence interval for the mean binary response $p_{\ast }=\Phi ({\eta }_{\ast })$ is