Sieve empirical likelihood ratio tests for nonparametric functions
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非参数函数的筛经验似然比检验
- 研究问题
- 在不指定随机误差完整分布时，如何检验非参数函数假设或条件估计方程，并获得不依赖干扰参数的渐近校准？
- 方法与思路
- 作者提出筛经验似然比检验，以条件估计方程替代完整误差分布设定，分析统计量渐近分布、检验功效并开展模拟。
- 材料陈述的贡献
- 摘要称统计量具有不依赖干扰参数的缩放卡方极限，延续Wilks现象，达到最优检验速率，并适应未知误差与异方差。
- 阅读关注点
- 应核对筛与带宽选择、条件估计方程的要求、缩放和自由度公式，以及Wilks现象、最优功效与异方差适应的条件。
- 阅读边界
- 这是arXiv预印本，同行评审与期刊接收未核实。摘要未列出正则条件及模拟数值，不能认为任意非参数模型均可直接使用。
原文摘要
Generalized likelihood ratio statistics have been proposed in Fan, Zhang and Zhang [Ann. Statist. 29 (2001) 153-193] as a generally applicable method for testing nonparametric hypotheses about nonparametric functions. The likelihood ratio statistics are constructed based on the assumption that the distributions of stochastic errors are in a certain parametric family. We extend their work to the case where the error distribution is completely unspecified via newly proposed sieve empirical likelihood ratio (SELR) tests. The approach is also applied to test conditional estimating equations on the distributions of stochastic errors. It is shown that the proposed SELR statistics follow asymptotically rescaled χ^2-distributions, with the scale constants and the degrees of freedom being independent of the nuisance parameters. This demonstrates that the Wilks phenomenon observed in Fan, Zhang and Zhang [Ann. Statist. 29 (2001) 153-193] continues to hold under more relaxed models and a larger class of techniques. The asymptotic power of the proposed test is also derived, which achieves the optimal rate for nonparametric hypothesis testing. The proposed approach has two advantages over the generalized likelihood ratio method: it requires one only to specify some conditional estimating equations rather than the entire distribution of the stochastic error, and the procedure adapts automatically to the unknown error distribution including heteroscedasticity. A simulation study is conducted to evaluate our proposed procedure empirically.