The GNSS integer ambiguities: estimation and validation

The quality of the solution can then be measured by the precision of the estimators.
However, the precision does not give information on the validity of the model, so that
the unbiasedness of the estimators cannot be guaranteed. It is therefore important to
use statistical tests in order to get information on the validity of the model.
2.5.1 Model testing
Statistical testing is possible if there is a null hypothesis, H0, that can be tested against
an alternative hypothesis, Ha. These hypotheses can be defined as:
H0 : E{y} = Ax; D{y} = Qy
Ha : E{y} = Ax + C∇; D{y} = Qy
(2.69)
where C is a known m × q-matrix that specifies the type of model error, and ∇ an
unknown q-vector. It is assumed that y is normally distributed, see appendix A.2.
H0 can be tested against Ha using the following test statistic:
T q = 1
q êT Q −1
y C C T Q −1
y QêQ −1
y C −1 C T Q −1
y ê (2.70)
The test statistic T q has a central F -distribution with q and ∞ degrees of freedom under
H0, and a non-central F -distribution under Ha:
H0 : T q ∼ F (q, ∞, 0); Ha : T q ∼ F (q, ∞, λ) (2.71)
with non-centrality parameter λ:
λ = ∇ T C T Q −1
y QêQ −1
y C∇ (2.72)
The test is then given by:
reject H0 if T q > Fα(q, ∞, 0), (2.73)
where α is a chosen value of the level of significance, also referred to as the false alarm
rate since it equals the probability of rejecting H0 when in fact it is true. Fα(q, ∞, 0) is
the critical value such that:
∞
α = fF (F |q, ∞, 0)dF (2.74)
Fα(q,∞,0)
where fF (F |q, ∞, 0) is the probability density function of F (q, ∞, 0), see appendix A.2.3.
The value of λ = λ0 can be computed once reference values are known for the level
of significance α = αq, and the detection power γ = γ0, which is the probability of
rejecting H0 when indeed Ha is true:
γ =
∞
Fα(q,∞,0)
fF (F |q, ∞, λ)dF (2.75)
Least-squares estimation and quality control 23