On the existence of minimum contrast estimates in binary response model

When random samples are drawn from a 3-parameter distribution with a shifted origin and the observations corresponding to each sample are binary, criteria for the existence of minimum contrast estimates are given. These criteria can be drived by a method, called the probability contents boundary analysis. The probabilities of the existence of maximum likelihood estimates and least squares estimates are evaluated, by simulation with 1000 replications, in the case where the underlying distribution is a 3-parameter lognormal distribution or a 3-parameter loglogistic distribution.