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arXiv 2109.09339stat.ME
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使用贝叶斯估计量提高列联表中指数估计的准确度

Improving the accuracy of estimating indexes in contingency tables using Bayesian estimators

Tomotaka Momozaki, Koji Cho, Tomoyuki Nakagawa, Sadao Tomizawa

备注：
19 pages, 6 figures
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AI总结：

针对列联表指数估计在样本量充足时偏差和均方误差较大的问题，提出基于Dirichlet先验的贝叶斯单元概率估计量，通过最小化MSE构建新指数估计量，在少样本下显著降低了偏差和MSE。

摘要：

在列联表分析中，人们感兴趣的是使用拟合优度检验来测试感兴趣的模型（例如独立性或对称性模型）是否成立。当模型为真的原假设被拒绝时，兴趣转向列联表的概率结构偏离该模型的程度。已经研究了许多指数来衡量偏离程度，例如用于独立性模型的 Yule 系数和 Cramér 系数，以及用于对称性模型的 Tomizawa 对称性指数。这些指数的推断是使用样本比例（即单元概率的估计）来进行的，但众所周知，在样本数量充足时，偏差和均方误差（MSE）值较大。为了解决这个问题，本研究提出了一种使用单元概率的贝叶斯估计量来估计指数的新估计量。假设单元概率的先验服从 Dirichlet 分布，我们通过将单元概率的后验均值代入指数来渐近评估 MSE 的值，并提出使用使该值最小化的 Dirichlet 超参数的指数估计量。数值实验表明，当每个单元的样本数较少时，所提出的方法在偏差和 MSE 方面具有比其他校正估计准确度方法更小的值。我们还表明，偏差和 MSE 的值小于使用均匀 Jeffreys 先验获得的值。

英文摘要：

In contingency table analysis, one is interested in testing whether a model of interest (e.g., the independent or symmetry model) holds using goodness-of-fit tests. When the null hypothesis where the model is true is rejected, the interest turns to the degree to which the probability structure of the contingency table deviates from the model. Many indexes have been studied to measure the degree of the departure, such as the Yule coefficient and Cramér coefficient for the independence model, and Tomizawa's symmetry index for the symmetry model. The inference of these indexes is performed using sample proportions, which are estimates of cell probabilities, but it is well-known that the bias and mean square error (MSE) values become large without a sufficient number of samples. To address the problem, this study proposes a new estimator for indexes using Bayesian estimators of cell probabilities. Assuming the Dirichlet distribution for the prior of cell probabilities, we asymptotically evaluate the value of MSE when plugging the posterior means of cell probabilities into the index, and propose an estimator of the index using the Dirichlet hyperparameter that minimizes the value. Numerical experiments show that when the number of samples per cell is small, the proposed method has smaller values of bias and MSE than other methods of correcting estimation accuracy. We also show that the values of bias and MSE are smaller than those obtained by using the uniform and Jeffreys priors.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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