Prediction Theory 1 Introduction 2 General Linear Mixed Model

5 Variances of Predictors
Let
then
P = (X ′ V −1 X) −
ˆb = PX ′ V −1 y
û = GZ ′ V −1 (y − XPX ′ V −1 y)
= GZ ′ V −1 Wy
for W = (I − XPX ′ V −1 ). From the results on generalized inverses of X,
and therefore,
The variance of the predictor is,
The covariance between ˆb and û is
XPX ′ V −1 X = X,
WX = (I − XPX ′ V −1 )X
= X − XPX ′ V −1 X
= X − X = 0.
V ar(û) = GZ ′ V −1 W(V ar(y))W ′ V −1 ZG
= GZ ′ V −1 WVW ′ V −1 ZG
Therefore, the total variance of the predictor is
= GZ ′ V −1 ZG − GZ ′ V −1 XPX ′ V −1 ZG.
Cov(ˆb, û) = PX ′ V −1 V ar(y)W ′ V −1 ZG
= PX ′ W ′ V −1 ZG
= 0 because X ′ W = 0
V ar(K ′ˆb + M ′ û) = K ′ PK + M ′ GZ ′ V −1 ZGM
−M ′ GZ ′ V −1 XPX ′ V −1 ZGM.
6 Variance of Prediction Error
The main results are
V ar(ˆb − b) = V ar(ˆb) + V ar(b) − Cov(ˆb, b) − Cov(b, ˆb)
= V ar(ˆb)
= P.
V ar(û − u) = V ar(û) + V ar(u) − Cov(û, u) − Cov(u, û),
6