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We study the $p$-generalized probit regression model, which is a generalizedlinear model for binary responses. It extends the standard probit model byreplacing its link function, the standard normal cdf, by a $p$-generalizednormal distribution for $p\in[1, \infty)$. The $p$-generalized normaldistributions \citep{Sub23} are of special interest in statistical modelingbecause they fit much more flexibly to data. Their tail behavior can becontrolled by choice of the parameter $p$, which influences the model'ssensitivity to outliers. Special cases include the Laplace, the Gaussian, andthe uniform distributions. We further show how the maximum likelihood estimatorfor $p$-generalized probit regression can be approximated efficiently up to afactor of $(1+\varepsilon)$ on large data by combining sketching techniqueswith importance subsampling to obtain a small data summary called coreset.

$p$-Generalized Probit Regression and Scalable Maximum Likelihood Estimation via Sketching and Coresets