Simultaneous confidence bands for a percentile hyper-plane with covariates constrained in a restricted range

A simultaneous confidence band provides useful information about the plausible range of the unknown regression model. In several recent papers, confidence bands have been used for various inferential purposes; see, for example, Liu et al. (2004, 2005, 2009, 2010, 2012, 2014), Liu and Hayter (2007) [1], Liu and Lin (2009), Piegorsch et al. (2005), Han et al. (2015) and Zhou (2017). The construction of simultaneous confidence bands has a long history, going back to Working and Hotelling (1929) [2], and is often difficult, when the region over which a confidence band is required is restricted and the number of predictor variables is greater than one. In this paper we have developed methods to construct simultaneous confidence bands for a percentile hyper-plane. These methods allow us to construct the bands over arbitrary intervals, either finite or infinite; they also allow us to construct the Type I and Type II bands. In order to get an idea of the efficiency of these bands, we have also developed the confidence sets for the bands and have compared the bands under the minimum volume confidence set optimality criterion. The proposed methods are illustrated by an example.