Comparison of hyperbolic and constant width simultaneous confidence bands in multiple linear regression under MVCS criterion

Abstract

Summary A simultaneous confidence band provides useful information on the plausible range of the unknown regression model, and different confidence bands can often be constructed for the same regression model. For a simple regression line, it is proposed in Liu and Hayter (2007) to use the area of the confidence set that corresponds to a confidence band as an optimality criterion in comparison of confidence bands; the smaller is the area of the confidence set, the better is the corresponding confidence band. This minimum area confidence set (MACS) criterion can clearly be generalized to the minimum volume confidence set (MVCS) criterion in study of confidence bands for a multiple linear regression model. In this paper the hyperbolic and constant width confidence bands for a multiple linear regression model over a particular ellipsoidal region of the predictor variables are compared under the MVCS criterion. It is observed that whether one band is better than the other depends on the magnitude of one particular angle that determines the size of the predictor variable region. When the angle and so the size of the predictor variable region is small, the constant width band is better than the hyperbolic band but only marginally. When the angle and so the size of the predictor variable region is large the hyperbolic band can be substantially better than the constant width band.

Simultaneous Confidence Bands for Linear Regression with Covariates Constrained in Intervals

“…The MVCS criterion for simultaneous confidence bands was introduced in Liu & Hayter (2007) and Liu et al (2009b). Since each 1− α level confidence band for x T b corresponds to a 1− α confidence set for the regression coefficients b (see e.g.…”

“…Naiman, 1984) and the minimum area/volume confidence set criterion (see e.g. Liu & Hayter, 2007; Liu et al. , 2009b).…”