Asymptotic Properties of Predicted Probabilities in Discrete Regression

The discrete outcome of a probability model is recordedas Y(i)=1 while otherwise Y(i)=0. y is the vector of observedoutcomes, p the corresponding probabilities, p^a consistent estimate of p, and residuals are defined ase = y - p^. Under quite general conditions, theasymptotic properties of p^ ensure that these residualshave zero mean and are uncorrelated with p^. Theseasymptotic results extend to the multinomial case. Theysupport certain measures of fit for discrete models.