The GNSS integer ambiguities: estimation and validation

IE
IEU
IU
LU
Figure 4.1: The set of relationships between the different classes of estimators: integer
equivariant estimators (IE), unbiased integer equivariant estimators (IEU), unbiased integer
estimators (IU), and linear unbiased estimators (LU).
By its definition, the class of IE estimators is larger than the class of admissible estimators
defined in section 3.1, and thus all admissible estimators are also integer equivariant.
Recall that the admissible integer estimators are unbiased, and therefore this class is
referred to here as the class of integer unbiased (IU) estimators.
The class of IE estimators is also larger than the class of linear unbiased (LU) estimators.
Let f T θ y be the linear estimator of θ = lT a a + l T b b for some fθ ∈ R m . This estimator is
unbiased if f T θ Aa + f T θ Bb = lT a a + l T b b, ∀a ∈ Rn , b ∈ R m with E{y} = Aa + Bb. This
gives:
fθ(y + Aa) = fθ(y) + l T a a ∀y ∈ R m , a ∈ R n
fθ(y + Bb) = fθ(y) + l T b b ∀y ∈ Rm , b ∈ R p (4.2)
Comparing this result with (4.1) shows indeed that the condition of linear unbiasedness
is more restrictive than that of integer equivariance, since the first condition must be
fulfilled for all a ∈ R n . Hence, the class of linear unbiased estimators is a subset of the
class of integer equivariant (IE) estimators, so that there must exist IE estimators that
are unbiased. Summarizing, this gives the following relationships: IEU = IE ∩ U = ∅,
LU ⊂ IEU, and IU ⊂ IEU (see figure 4.1).
The goal was to find an estimator that is ’best’ in a certain sense. The criterion of ’best’
that will be used is that the mean squared error (MSE) should be minimal. The reason
for this choice is that the MSE is a well-known probabilistic criterion to measure the
closeness of an estimator to its target value. Furthermore, the MSE-criterion is often
used as a measure for the quality of the float solution itself. With the MSE-criterion the
best integer equivariant (BIE) estimator, ˜ θ, is defined as:
˜θ = arg min
fθ∈IE E{(fθ(y) − θ) 2 } (4.3)
Note that the minimization is taken over all IE functions that satisfy the condition of
definition 4.1.1. The solution of this minimization problem is given by:
˜θ =
z∈ZnRp
(lT a z + lT b ζ)fy (y + A(a − z) + B(b − ζ)) dζ
fy (y + A(a − z) + B(b − ζ)) dζ
z∈ZnRp U
(4.4)
68 Best Integer Equivariant estimation