Linear Models with Exchangeably Distributed Errors

A generalization of the general linear model is considered. Let Y i = α + T i y + e i, i = 1, …, n, where the Y i are 1 × p observed random variables, the T i are 1 × (r − 1) constant vectors, and α and γ are unobserved constants. The exchangeable linear model (EGLM) occurs when we assume that the e i are unobserved and exchangeably, jointly normally distributed. The classical general linear model (CGLM) occurs when the e i are independent and identically distributed (iid). Optimal procedures for testing hypotheses about γ for the CGLM are also optimal for the EGLM. Optimal methods for testing hypotheses about α for the CGLM have size 1 for the EGLM. There is no sensible test that the errors are iid in the EGLM. Estimation is considered.