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Probit regression is based on the model $P(Y=1 | X) = \Phi(X'\beta)$, where $\Phi$ is the standard normal cumulative distribution function (cdf). Would it make sense to replace $\Phi$ by another cdf?

ADDED: If yes, is there any practical, historical, or theoretical reason that explains that the normal (and logit) distribution(s) have been favoured? Is the choice of another distribution implemented in statistical sofware (R or others)?

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Yes, it would. Two other distributions widely used for binary responses are the Logistic and the Gumbel but that does not preclude other choices.

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  • $\begingroup$ Thank you JohnK. I have edited my question to better reflect my questionings. In the case of the Gumbel distribution that you mention, would you have a reference (theoretical article or practical study)? $\endgroup$ – polmath Jul 27 '15 at 9:16
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    $\begingroup$ @polmath data.princeton.edu/wws509/notes/c3s7.html is a good summary of the procedures. $\endgroup$ – JohnK Jul 27 '15 at 9:30

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