A COMPARISON OF ALTERNATIVE APPROACHES TO SUPREMUM-NORM GOODNESS-OF-FIT TESTS WITH ESTIMATED PARAMETERS
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含估计参数的上确界范数拟合优度检验：不同方法的比较
- 研究问题
- 当复合原假设中的参数需要估计时，如何处理拟合优度检验的非标准极限分布，并比较不同校准方法的功效？
- 方法与思路
- 作者研究 Durbin 型临界值调整，讨论其与高斯过程极值理论及 Khmaladze 变换的联系，并开展模拟比较。
- 材料陈述的贡献
- 摘要称 Durbin 型调整具有较准确的检验尺寸；小样本模拟中，其功效高于变换法，部分情形下高于参数自助法。
- 阅读关注点
- 应核对参数模型与经验过程的条件、临界值近似的构造，以及功效比较是否在相近实际检验尺寸下进行。
- 阅读边界
- 仅据所给 Crossref 摘要，不能将模拟中的尺寸与功效表现推广至所有模型；具体设计与数值结果未说明。
原文摘要
Goodness-of-fit tests based on parametric empirical processes have nonstandard limiting distributions when the null hypothesis is composite — that is, when parameters of the null model are estimated. Several analytic solutions to this problem have been suggested, including the calculation of adjusted critical values for these nonstandard distributions and the transformation of the empirical process such that statistics based on the transformed process are asymptotically distribution-free. The approximation methods proposed by Durbin (1985, Journal of Applied Probability 22(1), 99–122) can be applied to conduct inference for tests based on supremum-norm statistics. The resulting tests have quite accurate size, a fact that has gone unrecognized in the econometrics literature. Some justification for this accuracy lies in the similar features that Durbin’s approximation methods share with the theory of extrema for Gaussian random fields and for Gauss-Markov processes. These adjustment techniques are also related to the transformation methodology proposed by Khmaladze (1981, Theory of Probability and Its Applications 26(2), 240–257) through the score function of the parametric model. Simulation experiments suggest that in small samples, Durbin-style adjustments result in tests that have higher power than tests based on transformed processes, and in some cases they have higher power than parametric bootstrap procedures.