Construction of Exact Simultaneous Confidence Bands for a Simple Linear Regression Model

Summary A simultaneous confidence band provides a variety of inferences on the unknown components of a regression model. There are several recent papers using confidence bands for various inferential purposes; see for example, Sun et al. (1999) , Spurrier (1999) , Al‐Saidy et al. (2003) , Liu et al. (2004) , Bhargava & Spurrier (2004) , Piegorsch et al. (2005) and Liu et al. (2007) . Construction of simultaneous confidence bands for a simple linear regression model has a rich history, going back to the work of Working & Hotelling (1929) . The purpose of this article is to consolidate the disparate modern literature on simultaneous confidence bands in linear regression, and to provide expressions for the construction of exact 1 −α level simultaneous confidence bands for a simple linear regression model of either one‐sided or two‐sided form. We center attention on the three most recognized shapes: hyperbolic, two‐segment, and three‐segment (which is also referred to as a trapezoidal shape and includes a constant‐width band as a special case). Some of these expressions have already appeared in the statistics literature, and some are newly derived in this article. The derivations typically involve a standard bivariate t random vector and its polar coordinate transformation.