  We study the $p$-generalized probit regression model, which is a generalized
linear model for binary responses. It extends the standard probit model by
replacing its link function, the standard normal cdf, by a $p$-generalized
normal distribution for $p\in[1, \infty)$. The $p$-generalized normal
distributions \citep{Sub23} are of special interest in statistical modeling
because they fit much more flexibly to data. Their tail behavior can be
controlled by choice of the parameter $p$, which influences the model's
sensitivity to outliers. Special cases include the Laplace, the Gaussian, and
the uniform distributions. We further show how the maximum likelihood estimator
for $p$-generalized probit regression can be approximated efficiently up to a
factor of $(1+\varepsilon)$ on large data by combining sketching techniques
with importance subsampling to obtain a small data summary called coreset.
