Linear prediction sufficiency in the misspecified linear model. Issue 21 (21st September 2020) - Markiewicz, Augustyn; Puntanen, Simo
Abstract: We consider the general linear model y = X β + ε supplemented with the new (future) unobservable random vector y *, coming from y * = X * β + ε *, where the expectation of y * is X * β and the covariance matrix of y * is known as well as the cross-covariance matrix between y * and y . We denote the supplemented model as M * . The misspecified supplemented model is denoted as M ¯ *, and the misspecification concerns the covariance part of the setup. Suppose that Fy is linearly sufficient for estimable parametric function X * β under M * . We give necessary and sufficient conditions that Fy continues to be linearly sufficient for X * β under the model M ¯ * . The corresponding properties regarding the linear prediction sufficiency with respect to ε * and y * are also studied.