Saddlepoint Approximations of the Distribution of the Person Parameter in the Two Parameter Logistic Model

Abstract

Large sample theory states the asymptotic normality of the maximum likelihood estimator of the person parameter in the two parameter logistic (2PL) model. In short tests, however, the assumption of normality can be grossly wrong. As a consequence, intended coverage rates may be exceeded and confidence intervals are revealed to be overly conservative. Methods belonging to the higher-order-theory, more specifically saddlepoint approximations, are a convenient way to deal with small-sample problems. Confidence bounds obtained by these means hold the approximate confidence level for a broad range of the person parameter. Moreover, an approximation to the exact distribution permits to compute median unbiased estimates (MUE) that are as likely to overestimate as to underestimate the true person parameter. Additionally, in small samples, these MUE are less mean-biased than the often-used maximum likelihood estimator.