Robust Wald-Type Tests of One-Sided Hypotheses in the Linear Model
线性模型中单侧假设的稳健 Wald 型检验AI 解读 ↓
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线性模型中单侧假设的稳健 Wald 型检验
- 研究问题
- 如何将稳健回归估计量用于单侧假设检验，使检验在长尾误差下维持检验尺寸与功效，并沿用常规临界值？
- 方法与思路
- 作者以渐近正态的稳健估计量构造 Wald 型统计量，在协方差具有特定比例结构时比较零分布和 Pitman 渐近效率。
- 材料陈述的贡献
- 摘要称在相应条件下稳健与常规统计量共享渐近零分布，估计量的相对效率可转化为检验效率，并报告长尾情形的模拟支持。
- 阅读关注点
- 应核对单侧假设集合、渐近正态性与协方差比例条件，以及非对称误差所需修改和有限样本检验尺寸的表现。
- 阅读边界
- 材料为 OpenAlex 索引摘要，稿件版本未注明；估计量符号及部分中心表述有提取缺失，需查原文；共享零分布结论不能无条件推广。
原文摘要
Let us consider the linear model Y = Xθ + E in the usual matrix notation, where the errors are iid. Our main objective is to develop robust Wald-type tests for a large class of hypotheses on θ; by robust we mean robustness in terms of size and power against long-tailed error distributions. First assume that the error distribution is symmetric about the origin. Let L and be the least squares and a robust estimator of θ. Assume that they are asymptotically normal about θ with covariance matrices σ 2(XtX)–1 and τ 2(XtX)–1, respectively. So could be an M estimator or a high breakdown point estimator. Robust Wald-type tests based on (denoted by RW) are studied here for testing a large class of one-sided hypotheses on θ. It is shown that the asymptotic null distribution of RW and that of the usual Wald-type statistic based on L (denoted by W) are the same. This is a useful result since the critical values and procedures for computing the p values for W are directly applicable to RW as well. A more important result is that the Pitman asymptotic efficiency of RW relative to W is (σ 2 /τ 2), which is precisely the asymptotic efficiency of relative to L . In other words, the efficiency robustness properties of relative to L translate to power robustness of RW relative to W. These results hold for asymmetric error distributions as well, with a minor modification. The general theory presented incorporates Wald-type statistics based on a large class of estimators that includes M estimators, bounded influence estimators, and high breakdown point estimators. The main requirement is that the estimator under consideration be asymptotically normal about 0; hence it does not explicitly require the errors to be iid or symmetric. If the asymptotic covariance of is not proportional to (XtX)–1, the asymptotic null distribution of RW and of W are unlikely to be the same. The results of a simulation study show that in realistic situations RW based on an M estimator is likely to have at least as much power as W, and more if the errors have long tails. A simple example illustrates the application of RW and its advantages over W.