In the analysis of two-way contingency tables, the measures for representing the degree of departure from independence, symmetry or asymmetry are often used. These measures in contingency tables are expressed as functions of the probability structure of the tables. Hence, the value of a measure is estimated. Plug-in estimators of measures with sample proportions are used to estimate the measures, but without sufficient sample size, the bias and mean squared error (MSE) of the estimators become large. This study proposes an estimator that can reduce the bias and MSE, even without a sufficient sample size, using the Bayesian estimators of cell probabilities. We asymptotically evaluate the MSE of the estimator of the measure plugging in the posterior means of the cell probabilities when the prior distribution of the cell probabilities is the Dirichlet distribution. As a result, we can derive the Dirichlet parameter that asymptotically minimizes the MSE of the estimator. Numerical experiments show that the proposed estimator has a smaller bias and MSE than the plug-in estimator with sample proportions, uniform prior, and Jeffreys prior. Another advantage of our approach is the construction of credible intervals for measures using Monte Carlo simulations.
翻译:在分析双向应急表时,经常使用代表脱离独立、对称或不对称程度的措施。应急表中的这些措施是作为各表概率结构的函数表示的。因此,对计量值进行了估计。使用具有抽样比例的计量的插图估测器来估计测量尺度,但没有足够样本大小,则测量器的偏差和平均平差(MSE)就会变大。本研究报告建议使用Bayesian细胞概率估计器,即使没有足够样本大小,也可以减少偏差和MSE。我们用Bayesian细胞概率估计器来表示这些措施的值。我们随机评估了测量器的测算器的数值值值值值值值值值值。当先前细胞概率分布为Drichlet分布时,细胞概率的偏差和平均正方差(MSE)就会变大。结果是,我们可以利用Drichlet参数尽可能减少估测器的MSE,即使没有足够样本大小,但利用Bayesian 估计器的测算仪显示,拟议的测算器的测算器在前的测算器上具有较可靠的偏差率性,而测算器的测算器在前的测算器上具有较小的测距。