Global Positioning System Reference
In-Depth Information
[
z
1
z
2
z
3
]
T
, which is a linear function of the random variables
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
2
,
according to Equations (4.235), (2.248), and (4.269). By using the covariance matrix
(4.88) and applying variance-covariance propagation, we find that the covariances
between the
z
i
are zero. Because the distribution of the
z
is multivariate normal, it
follows that the random variables
z
i
are stochastically independent. Since
(first group) and
v
T
Pv
is
a function of
z
3
only, it follows that
v
T
Pv
in (4.263), which is only a function of
z
2
,
and
∆
v
T
Pv
in (4.273) are stochastically independent. Thus, it is permissible to form
the following ratio of random variables:
∆
v
T
Pv
(n
1
−
∆
r)
∼
F
n
2
,n
1
−
r
(4.274)
v
T
Pv
∗
(n
2
)
w
hich has an
F
distribution.
Thus the fundamental test in sequential adjustment is based on the
F
distribution.
Th
e zero hypothesis states that the second group of observations does not distort
th
e adjustment, or that there is no indication that something is wrong with the second
gr
oup of observations. The alternative hypothesis states that there is an indication that
th
e second group of observations contains errors. The zero hypothesis is rejected, and
th
e alternative hypothesis is accepted if
[13
Lin
—
-1.
——
Sho
PgE
F<F
n
2
,n
1
−
r,
1
−α
/
2
(4.275)
F>F
n
2
,n
1
−
r,
α
/
2
(4.276)
Ta
ble 4.7 lists selected values from the
F
distribution as a function of the degrees
of
freedom and probability. The tabulation refers to the parameters as specified in
F
n
1
,n
2
,
0
.
05
.
[13
4.9.4 General Linear Hypothesis
Th
e general linear hypothesis deals with linear conditions between parameters. Non-
lin
ear conditions are first linearized. The basic idea is to test the change
v
T
Pv
for its
sta
tistical significance. Any of the three adjustment models can be used to carry out
∆
TABLE 4.7
Selected Values for
F
n
1
n
2
1
2
3
4
5
6.61
5.79
5.41
5.19
10
4.96
4.10
3.71
3.48
20
4.35
3.49
3.10
2.87
60
4.00
3.15
2.76
2.53
120
3.92
3.07
2.68
2.45
∞
3.84
3.00
2.60
2.37
