Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

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).