PROCESSING PSEUDORANGES AND CARRIER PHASES - GPS Satellite Surveying - page 291

N n − r w 2 ,

0 T − 1

2

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

z H a ∼ σ (7.198)

See Equation (4.272) for the corresponding expression for the zero hypothesis. The
matrix T has the same meaning as in Section 4.9.4, i.e.,

= A 2 N − 1

A 2 − 1 T (7.199)

1

The next step is to diagonalize the covariance matrix of Z H a and to compute the sum
of the squares of the transformed random variables. These newly formed random
variables have a unit variate normal distribution with a nonzero mean. According to
Section A.5.2, the sum of the squares has a noncentral chi-square distribution. Thus,

z H a Tz H a
σ

v T Pv
σ

∆

2
n 2 ,

= ∼ χ (7.200)

[29 λ

2
0

2
0

where the noncentrality parameter is

Lin
—
7 ——
No
*PgE

w 2 Tw 2
σ

λ = (7.201)

2
0

The reader is referred to the statistical literature, such as Koch (1988), for additional
details on noncentral distributions and their respective derivations. Finally, the ratio

v T Pv
v T Pv ∗

∆ n 1 − r

∼ F n 2 ,n 1 − r, λ (7.202)

[29

n 2

has a noncentral F distribution with noncentrality λ . If the test statistic computed

under the specifications of H 0 fulfills F ≤ F n 2 ,n 1 − r, α , then H 0 is accepted with a

type-I error of α . The alternative hypothesis H a can be separated from H 0 with the

power 1 − β ( α , λ ) . The type-II error is

F n 2 ,n 1 − r, 1 − α

β ( α , λ ) = F n 2 ,n 1 − r, λ dx (7.203)

0

Th e integration is taken over the noncentral F -distribution function from zero to the
va lue F n 2 ,n 1 − r, α , which is specified by the significance level .

Because the noncentrality is different for each alternative hypothesis according to
(7 .201), the type-II error

α

) also varies with H a . Rather than using the individual
ty pe-II errors to make decisions, Euler and Schaffrin (1990) propose using the ratio of
no ncentrality parameters. They designate the float solution as the common alternative
hypothesis H a , for all null hypotheses. In this case, the value w 2 in (7.195) is
w 2 =− A 2 x ∗ + 2

β ( α , λ

(7.204)

and the noncentrality parameter becomes