550 Inferences About Normal Linear Models
9.6.11.Let the independent random variablesY 1 ,...,Ynhave the joint pdf
L(α, β, σ^2 )=
(
1
2 πσ^2
)n/ 2
exp
{
−
1
2 σ^2
∑n
1
[yi−α−β(xi−x)]^2
}
,
where the given numbersx 1 ,x 2 ,...,xnare not all equal. LetH 0 :β=0(αand
σ^2 unspecified). It is desired to use a likelihood ratio test to testH 0 against all
possible alternatives. Find Λ and see whether the test can be based on a familiar
statistic.
Hint:In the notation of this section, show that
∑n
1
(Yi−αˆ)^2 =Q 3 +β̂^2
∑n
1
(xi−x)^2.
9.6.12.Using the notation of Section 9.2, assume that the meansμjsatisfy a linear
function ofj,namely,μj=c+d[j−(b+1)/2]. Let independent random samples
of size abe taken from thebnormal distributions having meansμ 1 ,μ 2 ,...,μb,
respectively, and common unknown varianceσ^2.
(a)Show that the maximum likelihood estimators ofcanddare, respectively,
cˆ=X..and
dˆ=
∑b
j=1[j−(b−1)/2](X.j−X..)
∑b
j=1[j−(b+1)/2]^2
.
(b)Show that
∑a
i=1
∑b
j=1
(Xij−X..)^2 =
∑a
i=1
∑b
j=1
[
Xij−X..−dˆ
(
j−
b+1
2
)] 2
+dˆ^2
∑b
j=1
a
(
j−
b+1
2
) 2
.
(c)Argue that the two terms in the right-hand member of part (b), once divided
byσ^2 , are independent random variables withχ^2 distributions provided that
d=0.
(d)WhatF-statistic would be used to test the equality of the means, that is,
H 0 :d=0?
9.6.13.Consider the discussion in Section 9.6.2.
(a)Show thatθˆ=ˆα 1 +βˆxc,whereˆαandβˆare the least squares estimators
derived in this section.
(b)Show that the vectorˆe=Y−ˆθis the vector of residuals; i.e., itsithentry is
eˆi, (9.6.7).