Linear prediction sufficiency in the misspecified linear model

Abstract

We consider the general linear model supplemented with the new (future) unobservable random vector coming from where the expectation of is and the covariance matrix of is known as well as the cross-covariance matrix between and . We denote the supplemented model as The misspecified supplemented model is denoted as and the misspecification concerns the covariance part of the setup. Suppose that is linearly sufficient for estimable parametric function under We give necessary and sufficient conditions that continues to be linearly sufficient for under the model The corresponding properties regarding the linear prediction sufficiency with respect to and are also studied.