Biehler Martin, Holling Heinz, Doebler Philipp
Westfälische Wilhelms-Universität Münster, Münster, Germany,
Psychometrika. 2015 Sep;80(3):665-88. doi: 10.1007/s11336-014-9405-1. Epub 2014 Apr 8.
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.
大样本理论阐述了两参数逻辑斯蒂（2PL）模型中个体参数最大似然估计量的渐近正态性。然而，在短测试中，正态性假设可能会严重错误。结果，可能会超过预期的覆盖率，并且置信区间被证明过于保守。属于高阶理论的方法，更具体地说是鞍点近似，是处理小样本问题的便捷方式。通过这些方法获得的置信界在个体参数的广泛范围内保持近似的置信水平。此外，对精确分布的近似允许计算中位数无偏估计（MUE），其高估和低估真实个体参数的可能性相同。此外，在小样本中，这些MUE比常用的最大似然估计量的均值偏差更小。
