Global Positioning System Reference
In-Depth Information
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where
0]
T
e
i
=
[0
···
010
···
(4.350)
de
notes an
n
1 vector containing 1 in position
i
and zeros elsewhere. The expected
va
lue and the covariance matrix are
×
E(
v
|
H
a
)
=−
Q
v
Pe
i
∇
i
(4.351)
ˆ
0
Q
v
Σ
v
|
H
a
=
Σ
v
= σ
(4.352)
It follows from Theorem 1 of Section 4.9.1 that
N
−
0
Q
v
[15
v
|
H
a
∼
Q
v
Pe
i
∇
i
,
σ
(4.353)
Since
P
is a diagonal matrix, the individual residuals are distributed as
Lin
—
0.8
——
Nor
*PgE
n
−
0
q
i
2
v
i
|
ˆ
H
a
∼
q
i
p
i
∇
i
,
σ
(4.354)
according to Theorem 2. Standardizing gives
n
−
,
1
H
a
σ
0
√
q
i
∼
v
i
|
ˆ
q
i
p
i
∇
i
σ
0
√
q
i
w
a
|
H
a
=
(4.355)
n
−
√
q
i
p
i
∇
i
σ
0
,
1
[15
=
or
n
−∇
i
p
i
√
q
i
σ
0
,
1
v
i
|
ˆ
H
a
σ
v
i
H
a
:
w
a
=
∼
(4.356)
Th
e zero hypothesis, which states that there is no blunder, is
v
i
|
ˆ
H
0
σ
v
i
H
0
:
w
0
=
∼
n(
0
,
1
)
(4.357)
The noncentrality parameter in (4.356), i.e., the mean of the noncentral normal dis-
tribution, is denoted by
δ
i
and is
δ
i
=
−∇
i
p
i
√
q
i
σ
0
=
−∇
i
√
r
i
σ
i
(4.358)
The parameter
δ
i
is a translation parameter of the normal distribution. The situation
is shown in Figure 4.10. The probability of committing an error of the first kind, i.e.,
of accepting the alternative hypothesis, equals the significance level
α
of the test
