ABSTRACT
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 (Morris, 1982;McCullagh and Nelder, 1989;Kotz et al., 2000; seeChapters 2 and3). In this chapter,Section 4.2 provides a general definition of exponential families and applies the definition to polytomous and continuous random vectors.Section 4.3 applies exponential families to IRT. InSection 4.4, special features of exponential families are applied to estimation problems encountered in IRT. Some concluding remarks concerning the role of exponential families are provided inSection 4.5.
