多维计算机化自适应测试中项目选择的渐近性
On the Asymptotics of Item Selection in Multidimensional Computerized Adaptive Testing
AI总结：
本研究针对多维计算机化自适应测试，提出加权A-最优项目选择规则，证明其渐近最优性，扩展至单项目施测场景，为MCAT的统计推断提供理论支撑，模拟验证了其有限样本表现。
AI中文摘要：
我们研究带有目标能力和干扰能力的多维计算机化自适应测试（MCAT）中基于费希尔信息的项目选择规则。尽管这类规则应用广泛，但针对自适应选择与潜特质估计耦合动力学的严格渐近理论仍较为有限。我们考虑一种加权A-最优准则，该准则在控制干扰维度精度的同时优先考虑目标能力。针对多维两参数逻辑模型，我们首先在项目类型复用机制下建立渐近最优性：结合最大似然估计量，加权A-最优规则可实现最小的渐近加权均方误差。随后我们证明最大似然估计量的一致性与渐近正态性，并通过极限费希尔信息矩阵表达其渐近加权均方误差。我们还将分析扩展至每个项目最多施测一次的实际场景。这些结果为MCAT中的正态近似及基于信息的标准误提供了依据。一项模拟研究检验了实际测试长度下的有限样本表现，显示出良好的加权均方误差，且项目选择模式与预测的极限行为一致。
英文摘要：
We study Fisher-information-based item-selection rules for multidimensional computerized adaptive testing (MCAT) with intentional and nuisance abilities. Although such rules are widely used, rigorous asymptotic theory for the coupled dynamics of adaptive selection and latent-trait estimation remains limited. We consider a weighted A-optimality criterion that prioritizes intentional abilities while controlling accuracy in nuisance dimensions. For the multidimensional two-parameter logistic model, we first establish asymptotic optimality under an item-type reuse regime: paired with the maximum likelihood estimator, the weighted A-optimal rule attains the minimum asymptotic weighted mean squared error. We then prove consistency and asymptotic normality of the maximum likelihood estimator and express its asymptotic weighted mean squared error through a limiting Fisher information matrix. We also extend the analysis to the operational setting in which each item is administered at most once. These results justify normal approximations and information-based standard errors in MCAT. A simulation study examines finite-sample behavior at practical test lengths and shows favorable weighted mean squared error, with item-selection patterns consistent with the predicted limiting behavior.
