ABSTRACT
This chapter provides a general definition of exponential families and applies the definition to polytomous and continuous random vectors. Exponential families are families of probability distributions characterized by a sufficient statistic. In item response model, their treatment emphasizes loglinear models for polytomous random variables and normal probability models for continuous random variables. Generally, exponential families are widely applied in statistical theory, for they include beta distributions, Dirichlet distributions, exponential distributions, gamma distributions, normal distributions, negative binomial distributions, Poisson distributions, and Weibull distributions. Exponential families are often used to construct families of distributions which involve mixtures. Such families are important in description of latent structure models used in item response theory.
