Pearson-Type Goodness-of-Fit Tests: The Univariate Case
Pearson型拟合优度检验：单变量情形AI 解读 ↓
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Pearson型拟合优度检验：单变量情形
- 研究问题
- 对未删失或随机删失的单变量数据，如何构造适用于简单和复合零假设、形式简洁的卡方拟合优度检验？
- 方法与思路
- 作者修改经验过程，使其弱极限为布朗运动，并在复合零假设下考虑原始数据最大似然及最小卡方估计。
- 材料陈述的贡献
- 摘要称检验相较通常形式增加一个自由度，可构造仅使用一个分组的检验，并给出删失情形的例子和实证展示。
- 阅读关注点
- 应查阅原PDF确认缺失的期望频数公式，核对删失机制、分组规则、参数估计对极限分布的影响及模拟结果。
- 阅读边界
- 索引摘要未注明版本，期望频数公式缺失，不能重构统计量；对删失分布的陈述也不足以排除其他删失假设。
原文摘要
Chi-squared tests are proposed for testing goodness of fit when the data may or may not be subject to random censoring. Both the simple and compositive null hypotheses are considered. For the simple null hypothesis the resulting test statistic in the uncensored case differs from Pearson's statistic only in that the expected frequency for cell Aj is taken to be , where is the empirical distribution function, instead of n ∫ A j d F o ; this results in k df instead of the usual k − 1. The theoretical justification lies in the behavior of the corresponding modified version of the usual empirical process: It converges weakly to a Brownian motion process instead of a tied-down Brownian motion. Under censoring, the tests require no restrictive assumptions on the censoring distribution and the statistic has the same simple form. For the composite null hypothesis the tests' development is based on the weak convergence of the modified empirical process when parameters are estimated. The raw-data maximum likelihood estimator and the minimum chi-squared estimator are considered in estimating unknown parameters. In both cases the form of the test statistic is the same as the usual chi-squared statistic, but again it has an additional degree of freedom. The additional degree of freedom results in simple tests based on only one cell. A number of examples are presented; the tests for the negative exponential, the Weibull, the Pareto, and the one-parameter gamma families are the simplest available tests under random censoring. Results of a simulation study and an analysis of a real data set are presented.