Multiple Comparisons for a Large Number of Parameters

When there are many parameters of interest (finitely large or infinite), standard multiple comparison procedures for a finite number of parameters (called discrete‐domain approaches) may lead to a simultaneous confidence region (SCR) too conservative to be useful. Such cases often arise in locating disease genes, detecting changes in image data and examining shapes and patterns in growth curves; or generally, in quantifying uncertainty in an estimate of a regression function (as one entity). In these cases, procedures designed for a continuous domain must be used. Scheffe's method is a classical example of continuous‐domain approaches. It provides an SCR for a regression function when errors are iid Gaussian and the predictor space is unconstrained, i.e. the domain of interest is the q dimensional Euclidean space. In practice, however, functions defined on finite intervals or other constrained domains are often of interest and data may not be Gaussian. Thus, Scheffe's SCR becomes either too conservative or inadequate. In this paper, we introduce and survey a modern‐type continuous‐domain approach, and explore a connection between some discrete‐ and continuous‐domain multiple comparison procedures. We show that, in some cases, even for a small number of parameters, it is still better to use a continuous‐domain multiple comparison procedure. The main ideas behind the continuous‐domain procedures are shown. A new procedure for comparing a finite number of contrasts about k regression curves is developed. Relevant software is provided.