is
a linear combination of
the
y/s.
Thus
we
have
for some constants
a
1
,
a
2
,
..•
,a
•.
Since the
y;'s
are assumed to be observed values of independent normal
variates, it follows by (6.6.
7)
that the sampling distribution of
a
is
normal.
It
can
be
shown that £(&)=ex, and (5.5.7) gives
var(&)=
"f.a'f
var(Y;) =
a
2
c
where
c
=
"f.a'f.
It
follows
that
&ex
Z
==.
CT:

N(O,
I).
v
aic
(13.2.6)
If
a
2
is
known,.inferences about
ex
are based
on
(13.2.
6)
. To test
H:
ex=
ex
0
,
we
compute the observed value of
Z
when
ex=
ex
0
,
and then find
SL=
P{
IZI
IZob•I}
=
P{xf1>
z;b.}·
It
can be shown that the likelihood ratio statistic for testing
H:
ex=
exa
is
Z
2
,
and
so the procedure
just
described
is
the likelihood ratio test.
To
construct a 95% confidence interval for
ex
when
a
is
known,
we
note
from Table
Bl
or
B2
that
P{ 
l.96:;;
Z:;;
l.96} = 0.95.
Substituting for
Z
and
solving gives
a

1.96#c
$ex
$a
+
l.96#c
as the 95% confidence interval. This
is
also a 5% significance interval:
it consists of
all
parameter values
ex
0
such that a likelihood ratio test of
H:
cx
=
ex
0
gives SL
0.05.
It
is
also a likelihood
or
maximum likelihood
interval for
ex
.
I
13.2
. Statistical Methods
203
Inferences for
o:,
/J,
...
:
<J
2
Unknown
Usually
a
1
is unknown,
and
is
estimated by
s
2
as defined in (13.2.5). By
(13.2.4)
we
have
Since
"f.er
is
distributed independently of&, it follows
that
v
is
distributed
independently of
Z
in
(I
3.2.6).
When
a
2
is
unknown,
we
consider the quantity
a

ex
T=
Fc
which
we
get by replacing
rJ
2
by
s
2
in (13.2.6). Note
that
where
Z

N(O,
I)
and
V
xfnq»
independently of
Z .
It follows by
(6
.
10
.
5)
that
T
has Student's distribution with
n

q
degrees of freedom:
(13.2.7)
Note that
T
has the same degrees of freedom as the variance estimate
s
2
.
To test
H:
ex=
ex
0
,
we
compute the observed value of
T
when
ex=
ex
0
,
and
then
use
Table BJ to find
SL=
P{lt(nq)'
17;,bsl}.
Alternatively,
we
have
SL=
P{tfnqJ
r;bs}
=
P{Fi
.•
q
r;b.}
by
(6.10.7), which can be evaluated from Table
BS
for the
F
distribution.
It
can
be
shown that the likelihood ratio statistic for testing
H:
ex=
ex
0
is
an
increasing function of
T
2
,
and hence the procedure
just
described
is
the
likelihood ratio test.
To construct a 95% confidence interval for
ex,
we
use Table
B3
to
find the
value
t
such that
P{
t
:s;
t<nqJ
St}
=
0.95.
Substituting from (13.2.7)
and
solving gives
&tFc
:s;ex:.:;&
+
tFc
as the 95% confidence interval. This is also a
5%
significance interval
and
a
maximum likelihood interval for
oc.
204
13
. Analysis
of
Normal Measurements
Inferences for
a
It
can be argued that, when the parameters
a,
p,
..
.
are unknown, the residual
sum of squares
r.e?
carries all of the information from the
y/s
concerning
a.
Inferences about
a
will therefore
be
based on the marginal distribution of
r.ef.
By
(13
.
2.4)
we
have
By
(6.9.1), the p.d.f. of
Vis
f(v)
=
k.vl•/2)
te
v/2
for
v
>
0,
where
v
=
n

q
and
kv
is
a constant.
If
we
now change variables using
(6.1.11),
we
find that the p.d.f. of
'f.ef
is
f(v)·

=k
vM2J

le
 •/
2,_
I
dv
I
1
dr.ef
v
a2
[
vs2J(v/2) l
{
vs2}
1
=k

exp

·
•
al
2
a2
a2
Based on this distribution, the log likelihood function of
a
is
vs
2
/(a)=

v
log
a
 
for
a>
0.
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 The Land, Maximum likelihood, Likelihood function