Jiayang Sun (Case Western Reserve University, USA)

Multiple Comparisons for Infinite Number of Parameters

When the number of parameters of interest is infinite or extremely large, such as, in the case of quantifying the uncertainty in the estimate of a regression curve or a response surface or a map or an image, itself, the standard multiple comparison methods for a finite number of parameters often lead to an infinite confidence bound or a test too conservative to be useful. In these cases, the methods designed for a continuous domain must be used. The Scheffe's method is a classical approach for such a purpose. It provides a simultaneous confidence bound for a regression function when errors are Gaussian, independent and homoscedastic, and the predictor space is unconstrained, i.e. the domain of interest is the whole q dimensional Euclidean space. In practice, we are often interested in functions defined on an interval or other restricted domains and i in other more general cases than the Gaussian. Thus the Scheffe's bound is also too conservative or inadequate in these cases and there have been attempts to provide good informative bounds in many important applications. In this talk, I'll introduce some modern techniques for simultaneous inferences and compare them with classical ones and others. Applications include simultaneous confidence bands for linear regression and nonparametric regression with homoscedastic, and heteroscedastic errors, growth and response curves with structured covariance matrices, and generalized linear models. Some "tricks" for these various models will be shown, real data examples and new (free) softwares will be provided.