Solution
= Solution
Conditionally on $x_i$,
$$
\mathbb P(Z_i=1\mid x_i)
=\Phi\left(\frac{\mu+x_i^T\beta-\tau}{\sigma}\right).
$$
Thus this is a probit regression with linear predictor
$$
\eta_i=\alpha+x_i^T\gamma,
\qquad \alpha=\frac{\mu-\tau}{\sigma},
\qquad \gamma=\frac\beta\sigma.
$$
Fit $(\alpha,\gamma)$ by maximum likelihood for Bernoulli observations, then recover
$$
\widehat\mu=\tau+\sigma\widehat\alpha,
\qquad \widehat\beta=\sigma\widehat\gamma.
$$