The Reduction of the Width of Confidence Bands in Linear Regression

Abstract

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.

Key Words:

• Average width
• Curvature of band
• Simultaneous confidence band