Interval Estimation in Linear Regression When Both Variables are Subject to Error
两个变量均含误差时线性回归的区间估计AI 解读 ↓
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两个变量均含误差时线性回归的区间估计
- 研究问题
- 在观测变量均有误差、独立观测对服从共同协方差的二元正态分布时，如何分别构造截距和斜率的置信区间？
- 方法与思路
- 作者在可用工具变量的框架下，从方差比构造的联合置信区域取单参数范围，并比较斜率区间与常见的 t 方法。
- 材料陈述的贡献
- 摘要称，所得单参数区间至少达到给定置信水平；斜率区间与 t 方法的差异发生在后者给出不相连区间时。
- 阅读关注点
- 应核对工具变量的要求、联合区域到单参数区间的转换，以及斜率区间出现无限界限的具体情形和解释。
- 阅读边界
- 仅据版本未注明的 OpenAlex 索引摘要；关键置信区域公式与不等式缺失，需查原 PDF，不能重构公式或核验覆盖率证明。
原文摘要
Define the linear functional relationship with parameters, α, β, by μ = α + βv. This paper considers the problem of confidence interval estimation of α and β separately, based on a sample of independent pairs, (x i , y i ), i = 1, 2, ···, n, with Ex i = vi, Ey i = μ i . The pairs, (x i , y i ), are assumed to be independent (i ≠ i′), to have a common (unknown) covariance matrix Σ, and to follow the bivariate normal distribution. It is known that under these circumstances (providing instrumental variates are available) one can define a variance ratio with 2 and (n − 2) degrees of freedom, F 2,n–2 say, as a function of α, β, sample statistics and instrumental variates to yield a confidence region on (α, β) based on In this paper we show, for interval estimation of α only, that taking limits as (min α, max α) satisfying yields confidence at least (1–γ). A similar result holds for interval estimation of β only. In the latter case, the limits obtained differ from those derived by the well-known procedure of defining a “t” statistic as a function of β and sample statistics. The difference, however, occurs only when the “t” statistic would yield a pair of disjoint intervals as an interval estimate of β (generally an event of low probability); in this instance, the method of this paper will, in one subset of cases, yield the same limits as the “t” statistic procedure, and, otherwise, will yield infinite limits for β.