Prediction Theory 1 Introduction 2 General Linear Mixed Model

3.1 Best Predictor
The best predictor, for any type of model, requires knowledge of the distribution of the random
variables as well as the moments of that distribution. Then, the best predictor is the conditional
mean of the predictor given the data vector, i.e.
E(K ′ b + M ′ u|y)
which is unbiased and has the smallest mean squared error of all predictors (Cochran 1951).
The computational form of the predictor depends on the distribution of y. The computational
form could be linear or nonlinear. The word best means that the predictor has the smallest
mean squared error of all predictors of K ′ b + M ′ u.
3.2 Best Linear Predictor
The best predictor may be linear OR nonlinear. Nonlinear predictors are often difficult to
manipulate or to derive a feasible solution. The predictor could be restricted to class of linear
functions of y. Then, the distributional form of y does not need to be known, and only the first
(means) and second (variances) moments of y must be known. If the first moment is Xb and
the second moment is V ar(y) = V, then the best linear predictor is
where
E(K ′ b + M ′ u) = K ′ b + C ′ V −1 (y − Xb)
C ′
= Cov(K ′ b + M ′ u, y).
When y has a multivariate normal distribution, then the best linear predictor (BLP) is the
same as the best predictor (BP). The BLP has the smallest mean squared error of all linear
predictors of K ′ b + M ′ u.
3.3 Best Linear Unbiased Predictor
In general, the first moment of y, namely Xb, is not known, but V, the second moment, is
commonly assumed to be known. Then predictors can be restricted further to those that are
linear and also unbiased. The best linear unbiased predictor is
where
and C and V are as before.
K ′ˆb + C ′ V −1 (y − Xˆb)
ˆb = (X ′ V −1 X) − X ′ V −1 y,
This predictor is the same as the BLP except that ˆb has replaced b in the formula. Note
that ˆb is the GLS estimate of b. Of all linear, unbiased predictors, BLUP has the smallest
mean squared error. However, if y is not normally distributed, then nonlinear predictors of
K ′ b + M ′ u could potentially exist that have smaller mean squared error than BLUP.
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