K -Sample Anderson–Darling Tests
多样本 Anderson–Darling 检验AI 解读 ↓
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多样本 Anderson–Darling 检验
- 研究问题
- 如何在不过度限制分布形式的情况下检验多个样本是否来自相同分布，并处理离散数据、数值取整和样本量不平衡？
- 方法与思路
- 作者提出两种秩统计量，对各组与合并样本的经验分布差异作加权积分，推导连续和离散情形的渐近分布，并用模拟考察近似。
- 材料陈述的贡献
- 摘要称检验对所有备择一致，提供临界值表及一种统计量的有限样本矩公式，并讨论合并独立检验统计量的用途。
- 阅读关注点
- 应核对两种经验分布定义、并列值处理及标准化方式，并查看小样本、取整程度和样本不平衡如何影响临界值近似。
- 阅读边界
- 仅据版本未注明的 OpenAlex 索引摘要，不能推断刊本变化。小样本近似效果来自所述模拟，不能保证任意离散分布下表现，也不能以未拒绝证明同分布。
原文摘要
Two k-sample versions of an Anderson–Darling rank statistic are proposed for testing the homogeneity of samples. Their asymptotic null distributions are derived for the continuous as well as the discrete case. In the continuous case the asymptotic distributions coincide with the (k – 1)-fold convolution of the asymptotic distribution for the Anderson–Darling one-sample statistic. The quality of this large sample approximation is investigated for small samples through Monte Carlo simulation. This is done for both versions of the statistic under various degrees of data rounding and sample size imbalances. Tables for carrying out these tests are provided, and their usage in combining independent one- or k-sample Anderson–Darling tests is pointed out. The test statistics are essentially based on a doubly weighted sum of integrated squared differences between the empirical distribution functions of the individual samples and that of the pooled sample. One weighting adjusts for the possibly different sample sizes, and the other is inside the integration placing more weight on tail differences of the compared distributions. The two versions differ mainly in the definition of the empirical distribution function. These tests are consistent against all alternatives. The use of these tests is two-fold: (a) in a one-way analysis of variance to establish differences in the sampled populations without making any restrictive parametric assumptions or (b) to justify the pooling of separate samples for increased sample size and power in further analyses. Exact finite sample mean and variance formulas for one of the two statistics are derived in the continuous case. It appears that the asymptotic standardized percentiles serve well as approximate critical points of the appropriately standardized statistics for individual sample sizes as low as 5. The application of the tests is illustrated with an example. Because of the convolution nature of the asymptotic distribution, a further use of these critical points is possible in combining independent Anderson–Darling tests by simply adding their test statistics.