Collaborative Research: Using Prior Kurtosis Information to Improve Confidence Intervals for Standard Deviations

本研究将应用Meta分析和Theil-Goldberger混合估计方法来改善样本方差的标准误差。这项研究将展示如何使用一种新的元分析峰度估计器来组合以前研究中的峰度信息，并且该估计器比标准的元分析峰度估计器的偏差更小。这项研究还将展示如何应用Theil-Goldberger方法将偏分析峰度估计与样本峰度估计混合，以获得样本方差的渐近无分布标准误差。然后，无分布的渐近标准误差将用于标准偏差的三个基本可信区间：1)单个标准偏差的可信区间，2)独立样本设计中两个标准偏差比率的可信区间，以及3)配对样本设计中两个标准偏差比率的可信区间。这项研究将检验三个基本可信区间在几个样本量和广泛的实际分布范围内的小样本覆盖概率。标准差或标准差比率的可信区间可以用来回答心理计量学、行为遗传学和质量控制中的基本问题。目前可用的方法基于不切实际的假设，如正态或等峰度，如果这些假设以微妙的方式被违反，使用标准诊断工具将很难检测到，那么这些方法的性能可能会很差。这项研究的结果将为科学家提供一套新的统计工具，可用于评估广泛应用中的可变性，并有望在现实条件下表现良好。

This research will apply meta-analysis and the Theil-Goldberger mixed estimation method to improve the standard error of a sample variance. The study will show how kurtosis information from previous studies can be combined using a new meta-analytic kurtosis estimator and that this estimator is less biased than the standard meta-analytic kurtosis estimator. The study also will show how to apply the Theil-Goldberger method to mix a meta-analytic kurtosis estimate with a sample kurtosis estimate to obtain an asymptotic distribution-free standard error of a sample variance. The asymptotic distribution-free standard error will then be used in three basic confidence intervals for standard deviations: 1) a confidence interval for a single standard deviation, 2) a confidence interval for a ratio of two standard deviations in independent-samples designs, and 3) a confidence interval for a ratio of two standard deviations in paired-samples designs. This research will examine the small-sample coverage probabilities of the three basic confidence intervals for several sample sizes and a wide range of realistic distributions.Confidence intervals for a standard deviation or a ratio of standard deviations can be used to answer fundamental questions in psychometrics, behavior genetics, and quality control. Currently available methods are based on unrealistic assumptions, such as normality or equal kurtosis, and can perform poorly if these assumptions are violated in subtle ways that would be difficult to detect using standard diagnostic tools. The results of this study will provide scientists with a new set of statistical tools that can be used to assess variability in a wide range of applications and can be expected to perform well under realistic conditions.