Smoothness of Parameterizations of Discrete Distributions

This chapter gives a detailed discussion of smooth parameterizations of discrete distributions. Smoothness essentially means that the function linking the parameters to the distributions may be well approximated locally by a linear transformation. This is a desirable property both in the asymptotic inference under the assumption of such a model and in the interpretation of parameter values. The concepts of parameters, parameterizations, and their smoothness are applied not only to probability but also to general distributions associating positive real values with the cells of the sample space. Properties of parameterizations of such distributions are simpler to study, but also a complete discussion is given for probability distributions. If a family of distributions possesses a smooth parameterization, then it is a smooth manifold, and properties of the smooth parameterization imply the dimension of the manifold. Various log-linear parameterizations are described in detail. To prepare the smoothness results to be presented in the next chapter, probability and general distributions are parameterized as exponential families, and the special roles of normalization and of the overall effect in parameterizing probability distributions are worked out.