The Reduction of the Width of Confidence Bands in Linear Regression
缩小线性回归置信带宽度的方法AI 解读 ↓
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缩小线性回归置信带宽度的方法
- 研究问题
- 在线性回归中，除改变置信带形状和限制自变量范围外，调整置信带曲率能否进一步降低平均宽度？
- 方法与思路
- 作者提出调整曲率的方案，给出结合形状、范围限制与曲率的定理，并对若干带形进行数值和图形比较。
- 材料陈述的贡献
- 摘要称仅限制自变量范围未必获得最小平均宽度，联合调整曲率能够改善结果，且梯形置信带尤其体现这一点。
- 阅读关注点
- 应核对置信带曲率的数学定义、同时覆盖概率、平均宽度的计算区域，以及定理和数值比较的适用条件。
- 阅读边界
- 所给索引摘要未注明版本，未给出置信带构造和缩窄幅度，不能声称普遍最优，也不能推断刊本变化。
原文摘要
When constructing a confidence band in the linear regression model, there have been various efforts made to reduce the width of the band. One of them is to change the shape of a confidence band. Another effort is to restrict the region of the independent variables. This article proposes another possibility to narrow down the width of a band, which is to change the curvature of the band. Following that a theorem is given to combine these devices. Numerical comparisons are carried out to see how the improvement due to the restriction of the independent variable works depending on the value of the curvature and the size of the restriction region for a couple of given band shapes. The results are shown by graphs. The following facts are observed. An improvement by restricting the region of the independent variables without changing the curvature of the band, which is my predecessors' method, does not bring the minimum average width. We must adjust the curvature at the same time to obtain the best band. Moreover, for a certain size of the restriction region, the average band width can be considerably reduced by choosing a suitable curvature of the band, even if the restriction is not taken into consideration. Especially when the shape of the confidence band is trapezoidal, the restriction of the independent variables may not work successfully for the purpose of reducing the band width, whereas the mere change of the curvature does. Provided we choose a suitable value of the curvature, the choice of the band shape does not affect the reduction of the band width.