Computational aspects of probit model

Sometimes the maximum likelihood estimation procedure for the probit model fails. There may be two reasons: the maximum likelihood estimate (MLE) just does not exist or computer overflow error occurs during the computation of the cumulative distribution function (cdf). For example, the approximation explosive effect due to an inaccurate computation of the cdf for a large value of the argument occurs in a popular statistical package S-plus. The goal of the paper is to provide remedies for these two abnormalities. First, despite the availability of a criterion for the MLE existence, expressed in terms of a separation plane in the covariate space, there are no constructive criteria to verify whether such a separation exists. We develop constructive criteria for the MLE existence that are valid also for other link functions. Second, to avoid the overflow problem we suggest approximate formulae for the log-likelihood function and its derivatives in the case of possible large value of the argument. Standard algorithms of the log-likelihood maximization like Newton-Raphson or Fisher Scoring are very sensitive to large values of the linear predictor, particularly outliers. Five algorithms are compared by the time to converge and reliability via statistical simulations. The corrected algorithms, based on the approximate formulae are more reliable and almost as fast as the standard ones.