Shorter Confidence Bands in Linear Regression
线性回归中更窄的置信带AI 解读 ↓
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AI 根据所提供的论文材料归类：摘要讨论一般线性回归置信带与变量约束，未提供对应所列领域细分的直接证据。
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线性回归中更窄的置信带
- 研究问题
- 当解释变量具有已知符号或多项式结构约束时，如何利用这些信息构造比通常Scheffé方法更窄的回归置信带？
- 方法与思路
- 作者针对解释变量符号已知的情形，推导多重比较程序置信概率的非平凡下界，并提供所需分布的分位点表。
- 材料陈述的贡献
- 摘要称该结果为较不保守的多重比较程序提供依据，并给出涉及两个、三个或四个参数线性组合的分位点表。
- 阅读关注点
- 应核对符号约束如何进入置信概率下界、置信带覆盖的变量范围，以及与Scheffé方法同时适用的条件。
- 阅读边界
- 仅据版本未注明的OpenAlex索引摘要；摘要未给出下界公式或缩窄幅度，不能将其比较结论扩展至任意回归问题。
原文摘要
In many linear regression problems, the values of the independent variable or variables may be subject to certain constraints. For example, the independent variables may necessarily be positive; as another example, the variables may not only all be positive but are powers of a single variable (e.g., polynomial regression on time). Previous writers considering the problem of obtaining confidence bands on a regression function for all values of the independent variable have not utilized such constraints; the usual basis for such bands has been the multiple comparison procedure of Scheffé which places no constraints at all upon the independent variables. Any procedure utilizing constraints will necessarily yield a uniform improvement over the method of Scheffé' (assuming both methods are applicable) in the sense of yielding narrower bands for a given confidence probability. In the present paper a non-trivial lower bound is obtained for the confidence probability associated with a multiple comparison procedure appropriate to the case where it can be assumed that each independent variable must be of specified sign; this includes, as a subclass, polynomial regression on a non-negative independent variable. This result gives a basis for a multiple comparison procedure less conservative than that of Scheffé when both are applicable. Implementation of the procedure requires the percentage points of a heretofore untabulated distribution. Tables of percentage points of this distribution appropriate to linear combinations of two, three, or four parameters are presented.