Theory of Statistics - George Mason University

We now write the original model as
Now, for the transformations. Let
Y = δ1 T n + ɛ.
Z = HmXH T n ,
5.5 Applications 433
δ = Hmδ,
and
ɛ = HmɛH T n .
We first of all note that the transformations are all nonsingular and
Z = H1 T n + ɛ.
Next, we see because of the orthonormality of the Helmert matrices that
the distributions of δ and ɛ are the same as those of δ and ɛ and they are still
independent. Furthermore, the Zij are independent, and we have
and
Zi1
iid ∼ N(0, σ 2 a + σ 2 ), for i = 1, . . ., m
Zij iid ∼ N(0, σ 2 ), for i = 1, . . ., m; j = 2, . . ., n0.
To continue with the analysis, we follow the same steps as in Example 5.28,
and get the same decomposition of the “adjusted total sum of squares” as in
equation (5.94):
m n
i=1 j=1
(Zij − Z) 2 = SSA + SSE. (5.99)
Again, we get chi-squared distributions, but the distribution involving SSA
is not the same as in expression (5.95) for the fixed-effects model.
Forming
MSA = SSA/(m − 1)
and
we see that
and
MSE = SSE/(m(n − 1)),
E(MSA) = nσ 2 δ + σ 2 ɛ
E(MSE) = σ 2 ɛ.
Unbiased estimators of σ 2 δ and σ2 ɛ are therefore
Theory of Statistics c○2000–2013 James E. Gentle
s 2 δ = (MSA − MSE)/n (5.100)