Global Positioning System Reference
In-Depth Information
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w
2
Tw
2
σ
=
∆
v
T
Pv
σ
λ ≡
(7.205)
2
0
2
0
v
T
Pv
is the change of the sum of squares due to the constraint of the null
hy
pothesis.
Let the null hypothesis that causes the smallest change
where
∆
v
T
Pv
be denoted by
H
sm
.
Th
e change in the sum of the squares and the noncentrality are
∆
v
T
Pv
sm
and
∆
λ
sm
,
re
spectively. For any other null hypothesis we have
λ
j
>
λ
sm
.If
v
T
Pv
j
λ
sm
≥ λ
0
α
β
j
∆
λ
j
v
T
Pv
sm
=
,
β
sm
,
(7.206)
∆
then the two ambiguity sets comprising the null hypotheses
H
sm
and
H
j
are suffi-
ciently discernible. Both hypotheses are sufficiently different to be distinguishable
by means of their type-II errors. Because of its better compatibility with the float
solution, the ambiguity set of the
H
sm
hypothesis is kept, and the set comprising
H
j
is discarded.
Figure 7.21 shows the ratio
[29
Lin
—
0.9
——
Sho
PgE
β
j
)
as a function of the degree of freedom
and the number of conditions. Euler and Schaffrin (1990) recommend a ratio between
5 and 10, which reflects a relatively large
λ
0
(
α
,
β
sm
,
β
j
. Since
H
sm
is the
hypothesis with the least impact on the adjustment, i.e., the most compatible with
the float solution, it is desirable to have
β
sm
and a smaller
β
j
(recall that the type-II error equals
the probability of accepting the wrong null hypothesis). Observing more satellites
reduces the ratio for given type-II errors.
β
sm
>
[29
Figure 7.21
Discernibility ratio.
(Permission by Springer Verlag.)
