BAN (Best Asymptotically Normal) estimates and their corresponding chi-squared test statistics form the core of classical categorical data analysis. Historically, these estimates were developed for the multinomial distribution. This article describes the optimal properties of these classical estimators and the form of their chi-squared test statistics. It is also shown that these are special cases of BAN estimators and chi-squared test statistics from a more general family of discrete multivariate distributions (Sum Symmetric Power Series Distributions), which include multinomial, negative multinomial, and multivariate Poisson as special cases. These results are in turn generalized to the multivariate exponential family. BAN estimators and their test statistics take on familiar forms and properties as the classes of distributions are generalized. Estimation and hypothesis testing are described from the point of view of constraint equations and the more familiar freedom equations.