Asymptotic Properties of Predicted Probabilities in Discrete Regression

The discrete outcome of a probability model is recorded as Y(i)=1 while otherwise Y(i)=0. y is the vector of observed outcomes, p the corresponding probabilities, p^ a consistent estimate of p, and residuals are defined as e = y - p^. Under quite general conditions, the asymptotic properties of p^ ensure that these residuals have zero mean and are uncorrelated with p^. These asymptotic results extend to the multinomial case. They support certain measures of fit for discrete models.