Simultaneous Confidence Bounds in Multiple Regression Using Predictor Variable Constraints
利用预测变量约束构造多元回归同时置信界AI 解读 ↓
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利用预测变量约束构造多元回归同时置信界
- 研究问题
- 当预测变量只能取多面体凸集内的值时，如何利用这一范围约束构造具有指定覆盖概率、且更窄的同时置信界？
- 方法与思路
- 作者将覆盖概率关联到随机向量在多面锥上的投影范数，借助蒙特卡洛近似，并提供矩形约束下的投影算法。
- 材料陈述的贡献
- 摘要称，在所考察设计与区域中，相较给定保守界，单侧置信界改善较大，双侧改善较小，并给出近似临界值。
- 阅读关注点
- 应核对回归误差与设计条件、预测变量约束的来源、锥投影算法，以及模拟近似误差和比较界的适用范围。
- 阅读边界
- 仅据版本未注明的 OpenAlex 摘要；覆盖概率通过近似计算，宽度改善限于所列设计和约束区域，不能推断普遍优势。
原文摘要
This article presents a method for approximating the coverage probability of simultaneous confidence bounds for multiple regression functions when the vector of predictor variables is constrained to lie in a polyhedral convex set. The method is useful because it allows one to construct simultaneous confidence intervals with prescribed coverage probability for the regression function evaluated at various settings of the predictor variables, which are narrower than bounds obtained without using the predictor variable constraints. For a family of two-sided simultaneous confidence bounds that includes Scheffé-type and constant-width bounds, the probability of coverage is related to the distribution of the maximum Euclidean norm of the projection onto a polyhedral cone for a pair of random vectors with known joint distribution. An analogous relation holds for one-sided bounds. If an algorithm for computing the projection onto the cone is available, then these results enable one to use the Monte Carlo method to approximate the coverage probabilities of bounds. Particular attention is given to the case in which lower and upper bounds can be specified for each of the predictor variables so that the constraint region for bounding the regression function is rectangular. An efficient algorithm for calculating projections onto the appropriate cone for rectangular constraint regions, which facilitates coverage probability approximation of Schefffé-type bounds, is presented. This algorithm is used to calculate approximate critical points of Scheffé-type bounds for some specific designs and constraint regions and we show that substantial improvements for one-sided bounds and modest improvements for two-sided bounds over the conservative bounds of Casella and Strawderman (1980) may be obtained.