Theory of Statistics - George Mason University

432 5 Unbiased Point Estimation
You are to show this in Exercise 5.10. Compare this with the MLE of σ 2 in
Example 6.27 in Chapter 6.
The model in equation (5.88) is called the one-way AOV model. If the αi
in this model are assumed to be constants, it is called a “fixed-effects model”.
A fixed-effects model is also sometimes called “model I”. Now let’s consider a
variant of this called a “random-effects model” or “model II”, because the αi
in this model are assumed to be iid random variables.
Example 5.31 UMVUEs of the variances in the one-way randomeffects
AOV model
Consider the linear model
Yij = µ + δi + ɛij, i = 1, . . ., m; j = 1, . . ., n, (5.97)
where the δi are identically distributed with E(δi) = 0, V(δi) = σ2 δ , and
Cov(δi, δĩ ) = 0 for i = ĩ, and the ɛij are independent of the δi and are
identically distributed with with E(ɛij) = 0, V(ɛij) = σ2 ɛ, and Cov(ɛij, ɛĩ˜j ) = 0
for either i = ĩ or j = ˜j.
An important difference in the random-effects model and the fixed-effects
model is that in the random-effects model, we do not have independence of
the observables. We have
⎧
Cov(Yij, Yĩ˜j ) =
σ 2 δ + σ2 ɛ for i = ĩ, j = ˜j,
⎨
σ
⎩
2 δ for i = ĩ, j = ˜j,
0 for i = ĩ.
(5.98)
A model such as this may be appropriate when there are a large number
of possible treatments and m of them are chosen randomly and applied to
experimental units whose responses Yij are observed. While in the fixed-effects
model (5.88), we are interested in whether α1 = · · · = αm = 0, in the random-
effects model, we are interested in whether σ2 δ = 0, which would result in a
similar practical decision about the treatments.
In the model (5.97) the variance of each Yij is σ2 δ +σ2 ɛ, and our interest in
using the model is to make inference on the relative sizes of the components of
the variance σ2 δ and σ2 ɛ. The model is sometimes called a “variance components
model”.
Let us suppose now that δi iid ∼ N(0, σ2 δ ), where σ2 δ ≥ 0, and ɛij iid ∼ N(0, σ2 ɛ),
where as usual σ2 > 0. This will allow us to determine exact sampling distributions
of the relevant statistics.
We transform the model using Helmert matrices Hm and Hn as in equation
(5.82).
Let
⎡ ⎤ ⎡ ⎤ ⎡ ⎤
Y11 · · · Y1n δ1
ɛ11 · · · ɛ1n
⎢
Y =
. . ⎥ ⎢
⎣ . . ⎦ ; δ =
. ⎥ ⎢
⎣ . ⎦; and ɛ =
. . ⎥
⎣ . . ⎦.
Ym1 · · · Ymn
δm
Theory of Statistics c○2000–2013 James E. Gentle
ɛm1 · · · ɛmn