On the statistical analysis of grouped data: when Pearson χ2 and other divisible statistics are not goodness-of-fit
本文针对分组数据分析中常见的误区，提出统一框架处理可分割统计量（如皮尔逊χ²、似然比等），发现当分组数多且频数小时，多数检验可改进以提高功效，且无单一可分割统计量能用于拟合优度检验。
Abstract Thousands of experiments are analysed, and papers are published each year involving the statistical analysis of grouped data. While this area of statistics is often perceived–somewhat naively–as saturated, several misconceptions still affect everyday practice, and new frontiers have so far remained unexplored. Researchers must be aware of the limitations affecting their analyses and what new possibilities are at their hands. The article introduces a unifying approach to the analysis of divisible statistics–that includes Pearson’s χ2, the likelihood ratio, and spectral statistics, as special cases– when a statistician deals with a large number of bins/groups, thus leading to a large number of small or moderate frequencies. Performance of the tests is analysed against the class of contiguous (local) alternatives. Perhaps the most surprising result here is that, in this ‘sparse’ regime, most of the tests proposed in the literature can be modified to produce more powerful tests, and no single test based on a divisible statistic leads to a goodness-of-fit test. Distribution-free goodness-of-fit tests are also constructed.