Theory of integer equivariant estimation with application
to GNSS
P. J. G. Teunissen
Department of Mathematical Geodesy and Positioning, Delft University of Technology, Thijsseweg 11, 2629 JA Delft,
The Netherlands; e-mail: p.j.g.teunissen@citg.tudelft.nl; Tel.: +31-15-278-2558; Fax: + 31-15-278-3711
Received: 18 November 2002 / Accepted: 3 June 2003
Abstract. Carrier phase ambiguity res olution is the key
to high-precision global navigation satellite system
(GNSS) positioning and navigation. It applies to a
great variety of current and future models of GPS,
modernized GPS and Galileo. The so-called ‘fixed’
baseline estimator is known to be superior to its ‘float’
counterpart in the sense that its probability of being
close to the unknown but true baseline is larger than
that of the ‘float’ baseline, provided that the ambiguity
success rate is sufficiently close to its maximum value
of one. Although this is a strong result, the necessary
condition on the success rate does not make it hold
for all measurement scenarios. It is discussed whether
or not it is possible to take advantage of the integer
nature of the ambiguities so as to come up with a
baseline estimator that is always superior to both its
‘float’ and its ‘fixed’ counterparts. It is shown that
this is indeed possible, be it that the result comes at
the price of having to use a weaker performance
criterion. The main result of this work is a Gauss–
Markov-like theorem which introduces a new mini-
mum variance unbiased estimator that is always
superior to the well-known best linear unbiase d
(BLU) estimator of the Gauss–Markov theorem. This
result is made possible by introducing a new class
of estimators. This class of integer equivariant
estimators obeys the integer remove–restore principle
and is shown to be larger than the class of integer
estimators as well as larger than the class of linear
unbiased estimators. The minimum vari ance unbiased
estimator within this larger class is referred to as the
best integer equivariant (BIE) estimator. The theory
presented applies to any model of observation equa-
tions having both integer and real-valued parameters,
as well as for any probability density function the data
might have.
Keywords: Global navigation satellite system ambiguity
resolution – Integer equivariant estimation – Minimum
variance unbiased estimation
1 Introduction
Global navigation satellite system (GNSS) amb iguity
resolution is the process of resolving the unknown cycle
ambiguities of double-difference (DD) carrier phase
data. Its practical importance becomes clear when we
realize the great variety of current and future GNSS
models to which it applies. These models may differ
greatly in complexity and diversity. They range from
single-baseline models used for kinematic positioning to
multi-baseline models used as a tool for studying
geodynamic phenomena. The models may or may not
have the relative receiver–satellite geometry included.
They may also be discrimina ted as to whether the slave
receiver(s) is stationary or in motion. When in motion,
we solve for one or more trajectories, since with the
receiver–satellite geometry included, we will ha ve new
coordinate unknowns for each epoch. We may also
discriminate between the models as to whether or not
the differential atmospheric delays (ionosphere and
troposhere) are included as unknowns. In the case of
sufficiently short baselines they are usually excluded.
Apart from the current global positioning system
(GPS) models, carrier phase ambiguity resolution also
applies to the future modernized GPS and the future
European Galileo GNSS. An overview of GNSS mod-
els, together with their applications in surveying, navi-
gation, geodesy and geophysics, can be found in
textbooks such as Leick (1995), Parkinson and Spilker
(1996), Strang and Borre (1997), Teunissen and Kleus-
berg (1998), Hofmann-Wellenhof et al. (2001 ) and
Misra and Enge (2001).
Since carrier phase ambiguity resolution is the key to
high-precision GNSS positioning and navigation, the
availability of a theory of integer inference is a pre-
requisite for a proper handling and understanding of the
various intricate aspects of ambiguity resolution. The
usual approach to carrier phase ambiguity resolution is
to resolve the ambiguities once the probability of correct
Journal of Geodesy (2003) 77: 402–410
DOI 10.1007/s00190-003-0344-3
integer estimation, the ambiguity success rate, is suffi-
ciently close to one. When this happens we can show
that the so-called ‘fixed’ baseline estimator is superior to
its ‘float’ counterpart in the sense that its probability of
being close to the unknown but true baseline is larger
than that of the ‘float’ baseline. Although this is a strong
result, the down side of it is that it only holds true when
the success rate is sufficiently large. It is this observation
that formed the basis of our motiv ation for conducting
the present study. The question that will be an swered is
whether or not it is possible to take advantage of the
integer nature of the ambiguities so as to come up with a
baseline estimator that is always superior to both its
‘float’ and its ‘fixed’ counterparts. We will show that this
is indeed possible, be it that the result comes at the price
of having to use a weaker performance criterion. The
performance criterion chosen is the mean square error
(MSE). The reason for taking the MSE as the weaker
performance criterion is twofold. First, it is a well-
known probabilistic criterion for measuring the close-
ness of an estimator to its target value. Second, this
criterion is also often used as measure for the quality of
the ‘float’ solution itself.
Although the present study was motivated by the
problem of GNSS ambiguity resolution, the theory that
will be developed is of interest in its own right. It applies
to any model of observation equations having unknown
integer parameters as well as unknown real-valued
parameters. Our main result is a Gauss–Markov-like
theorem which introduces an estimator that is always
superior to the well-known best linear unbiased (BLU)
estimator of the Gauss–Markov theorem. The Gauss–
Markov theorem states that the minimum variance
unbiased estimator within the class of linear estimators is
given by the least-squares (LS) estimator. Our theorem
states that the minimum variance unbiased estimator
within the class of integer equivariant (IE) estimators is
given by the least mean-squared (LMS) estimator. This
estimator, referred to as the best integer equivariant
(BIE) estimator, is superior to the BLU estimator since
the class of linear unbiased estimators can be shown to be
a subset of the class of IE estimators. In the same sense it
can also be shown to be superior to integer estimators.
This contribution is organized as follows. In Sect. 2
we give a brief review of the present theory of integer
estimation. This includes the definition of the class of
admissible integer estimators. This class is taken as our
point of departure for introducing the class of integer
equivariant (IE) estimators in Sect. 3. Although this new
class is larger than the class of integer estimators, it has
been chosen such that its members still obey the integer
remove–restore principle. When estimating ambiguities
in case of GNSS, for instance, it seems reasonable to
require, when adding an arbitrary number of cycles to
the carrier phase data, that the solution of the integer
ambiguities gets shifted by the same integer amount. In
Sect. 3 we show that the class of linear unbiased esti-
mators is a subset of the class of IE estimators. We also
give a useful representation of IE estimators. This rep-
resentation reveals the structure of IE estimators and
easily allows us to devise our own IE estimator.
In Sect. 4 we use the MSE criterion to find the best
estimator within the IE class for any linear function of
both the integer as well as real-valued parameters of
the general GNSS model. We give an explicit expres-
sion for the BIE estimator. Although IE estimators
are not unbiased in general, we show in Sect. 5 that
the BIE estimator is unbiased. This implies that the
BIE estimator is identical to the BIE unbiased esti-
mator. This result gives rise to our Gauss–Markov-like
theorem stating the minimum vari ance unbiasedness
property of the BIE estimator. Although the BIE
estimator holds true for any probability density func-
tion the data might have, we also consider the special
case of normally distributed data. For this special case
it is shown that the BIE estimator of the baseline can
be obtained in a way which is very similar to the
three-step procedure of current methods of ambiguity
resolution, the only difference being that the integer
ambiguity estimator needs to be replaced by its BIE
counterpart.
2 Integer estimation
2.1 The GNSS model
As our point of departure we take the following syste m
of linear observation equations:
Efyg¼Aa þ Bb; a 2 Z
n
; b 2 R
p
ð1Þ
with Efg the mathematical expectation operator, y the
m vector of observables, a the n vecto r of unknown
integer parameters and b the p vector of unknown real-
valued parameters. The m ðn þ pÞ design matrix ð A ; BÞ
is assumed to be of full rank.
All the linear(ized) GNSS models can in principle be
cast in the above frame of observation equations. The
data vector y will then usually consist of the ‘observed
minus computed’ single- or dual-frequency DD phase
and/or pseudorange (code) observations accumulated
over all observation epochs. The entries of vector a are
then the DD carrier phase ambiguities, expressed in
units of cycles rather than range, while the entries of the
vector b will consist of the remaining unknown param-
eters, such as for instance baseline components (coor-
dinates) and possibly atmospheric delay parameters
(troposphere, ionosphere).
Although the theory that will be developed in this
contribution holds true for any application for which the
observation equations can be formulated as Eq. (1), we
will still refer to it as the GNSS model.
2.2 The three-step solution
The procedure which is usually followed for solving the
GNSS model can be divided into three steps. In the first
step we simply discard the integer constraints a 2 Z
n
and
perform a standard LS adjustment. As a result we obtain
the LS estimators of a and b as
403
^
aa ¼ð
AA
T
Q
1
y
AAÞ
1
AA
T
Q
1
y
y
^
bb ¼ð
BB
T
Q
1
y
BBÞ
1
BB
T
Q
1
y
y
ð2Þ
with Q
y
the vc matrix of the observables,
AA ¼ P
?
B
A,
BB ¼ P
?
A
B, and the tw o orthogonal projectors P
?
B
¼
I
m
BðB
T
Q
1
y
BÞ
1
B
T
Q
1
y
and P
?
A
¼ I
m
AðA
T
Q
1
y
AÞ
1
A
T
Q
1
y
. This solution is usually referred to as the ‘float’
solution.
In the second step the ‘float’ estimator
^
aa is used to
compute the corresponding integer estimator
aa 2 Z
n
.
This implies that a mapping S from the n-dimensional
space of reals to the n-dimensional space of integers is
introduced such that
aa ¼ Sð
^
aaÞ; S : R
n
7!Z
n
ð3Þ
This integer estimator is then used in the third and final
step to adjust the ‘float’ estimator
^
bb. As a result we
obtain the so-called ‘fixed’ estimator of b as
bb ¼
^
bb Q
^
bb
^
aa
Q
1
^
aa
ð
^
aa
aaÞð4Þ
in which Q
^
aa
denotes the vc matrix of
^
aa and Q
^
bb^aa
denotes
the covariance matrix of
^
bb and
^
aa. This ‘fixed’
estimator can alternatively be expressed as
bb ¼
ðB
T
Q
1
y
BÞ
1
B
T
Q
1
y
ðy A
aaÞ. Note that only two of the
three steps are needed if we are only interested in
obtaining an integer solution for a. In the case of GNSS,
however, we are particularly interested in the solution of
the third step as it contains the solution for the baseline
coordinates. All three steps are therefore needed in the
case of GNSS. In the following we will use the
terminology of GNSS and refer to
^
bb and
bb as,
respectively, the ‘float’ and ‘fixed’ baseline estimators.
The above three-step procedure is still ambiguous in
the sense that it leaves room for choosing the integer
map S. Different choices for S will lead to different
integer estimators
aa and thus also to different baseline
estimators
bb. We can therefore now think of construct-
ing integer maps which possess certain desirable prop-
erties.
2.3 A class of integer estimators
It will be clear that the map S will not be one-to-one due
to the discrete nature of Z
n
. Instead it will be a many-
to-one map. This implies that different real-valued
vectors will be mapped to one and the same integer
vector. We can therefore assig n a subset S
z
R
n
to each
integer vector z 2 Z
n
S
z
¼fx 2 R
n
j z ¼ SðxÞg; z 2 Z
n
ð5Þ
The subset S
z
contains all real-valued vectors that will be
mapped by S to the same integer vector z 2 Z
n
. This
subset is referred to as the pull-in region of z. It is the
region in which all vectors are pulled to the same integer
vector z.
Since the pull-i n regions define the integer estimator
completely, we can define classes of integer estimators by
imposing various conditions on the pull-in regions. One
such class was introduced by Teunissen (1999a) and is
referred to as the class of admissible integer estimators.
Definition 1: admissible integer estimators. The integer
estimator
aa ¼ Sð
^
aaÞ is said to be admissible if its pull-in
regions satisfy
1.
S
z2Z
n
S
z
¼ R
n
2. Intð S
z
1
Þ
T
IntðS
z
2
Þ¼;; 8z
1
; z
2
2 Z
n
; z
1
6¼ z
2
3. S
z
¼ z þ S
0
; 8z 2 Z
n
This definition is motivated as follows. Each one of the
above three conditions describes a property of which it
seems reasonable is possessed by an arbitrary integer
estimator. The first condition states that the pull-in
regions should not leave any gaps and the second that
they should not overlap. The absence of gaps is needed
in order to be able to map any ‘float’ solution
^
aa 2 R
n
to
Z
n
, while the absence of overlaps is needed to guarantee
that the ‘float’ solution is mapped to just one integer
vector. Note that we allow the pull-in regions to have
common boundaries. This is permitted if we assume to
have zero probability that
^
aa lies on one of the
boundaries. This will be the case when the probability
density function (PDF) of
^
aa is continuous.
The third and last condition of the de finition follows
from the requirement that Sðx þzÞ¼SðxÞþz;
8x 2 R
n
; z 2 Z
n
. This condition is also a reasonable one
to ask for. It states that when the ‘float’ solution
^
aa is
perturbed by z 2 Z
n
, the corresponding integer solution
is perturbed by the same amount. This property allows
us to apply the integer remove–restore technique:
Sð
^
aa zÞþz ¼ Sð
^
aaÞ. It therefore allows us to work with
the fractional parts of the entries of
^
aa, instead of with its
complete entries.
Using the pull-in regions, we can give an explicit
expression for the corresponding integer estimator
aa.It
reads
aa ¼
X
z2Z
n
zs
z
ð
^
aaÞð6Þ
with the indicator function s
z
ð
^
aaÞ¼1if
^
aa 2 S
z
and
s
z
ð
^
aaÞ¼0 otherwise. Note that the s
z
ð
^
aaÞ can be inter-
preted as weights, since
P
z2Z
n
s
z
ð
^
aaÞ¼1. The integer
estimator
aa is therefore equal to a weighted sum of
integer vectors with binary weights.
2.4 The performance of the baselin e estimators
As mentioned earlier the ‘fixed’ baseline estimator
bb
depends on the chosen integer estimator
aa. We may also
write
bb as
bb ¼
^
bbðaÞþQ
^
bb
^
aa
Q
1
^aa
ð
aa aÞð7Þ
with the conditional baseline estim ator
^
bbðaÞ¼
^
bb
Q
^
bb
^
aa
Q
1
^
aa
ð
^
aa aÞ. Since a is assumed known in the case
of
^
bbðaÞ, the conditional baseline estimator is the best
possible estimator of b. It is unbiased, it has a precision
which is better than that of
^
bb, and in the case that y is
normally distributed its PDF is also more peake d than
404
that of the ‘float’ estimator. The best we can hope for in
the case of
bb is therefore that
aa ¼ a. However, this
requires that the probability of correct integer estima-
tion, Pð
aa ¼ aÞ, equals one. Hence, of all admissible
integer estimators the preferred estimator is the one that
maximizes the probability of correct integer estimation.
It was shown by Teunissen (1999b) that in the case
of elliptically contoured distributions, the integer
least-squares (ILS) estimator is the preferred esti-
mator. Thus, if
aa
ILS
¼ arg min
z2Z
n
ð
^
aa zÞ
T
Q
1
^
aa
ð
^
aa zÞ,
then Pð
aa
ILS
¼ aÞPð
aa ¼ aÞ for any admissible estima-
tor
aa. And once the probability of correct integer
estimation is sufficiently close to one, we have
Pð
bb 2 E
b
ÞPð
^
bb 2 E
b
Þð8Þ
for any convex region E
b
R
p
symmetric with respect to
b. The usual approach taken with GNSS is therefore to
use the ILS estimator for computing the ‘fixed’ baseline
estimator, once it has been verified that Pð
aa
ILS
¼ aÞ is
sufficiently close to one, see (Teunissen 1993, 1995).
When valid, the inequality of Eq. (8) is a very strong
result. It states that the ‘fixed’ baseline estimator
bb has a
higher probability of being close to b than its ‘float’
counterpart. The down side of the inequality is, however,
that it is ony valid if Pð
aa ¼ aÞ is sufficiently close to one.
This being the case, we may wonder whether it would not
be possible to devise a baseline estimator which always
outperforms its ‘float’ counterpart. Such an approach
can, however, only be successful if we use a weaker per-
formance criterion than that of Eq. (8). Furthermore,
assuming that we will be successful in finding an optimal
estimator using this weaker criterion, then this new esti-
mator will always be better than its ‘float’ counterpart
only if we consider a class of estimators which encom-
passes the class of estimators in which the ‘float’ estimator
resides. This means that, as a start, we should at least
consider a class of estimators which is larger than the
above considered class of integer estimators.
3 Integer equivariant estimation
3.1 Class of IE estimators
We will now introduce a new class of estimators which is
larger than the previously defined class of integer
estimators. In order to be general enough, we consider
estimating an arbitrary linear function of the two types
of unknown parameters of the GNSS model of Eq. (1)
h ¼ l
T
a
a þl
T
b
b; l
a
2 R
n
; l
b
2 R
p
ð9Þ
Thus if l
b
¼ 0 then linear functions of the ambiguities
are estimated, whereas if l
a
¼ 0 then linear functions of
the baseline are estimated. Linear functions of both the
ambiguities and the baseline, such as carrier phases, are
estimated in the case that l
a
6¼ 0 and l
b
6¼ 0.
It seems reasonable that the estimator should at least
obey the integer remove–restore principle. When esti-
mating ambiguities in the case of GNSS, for instance,
when adding an arbitrary number of cycles to the carrier
phase data, we would like the solution of the integer
ambiguities to be shifted by the same integer amount.
For the estimator of h this would mean that adding Az to
y, with arbitrary z 2 Z
n
, must result in a shift of l
T
a
z.
Likewise, it seems reasonable to require of the estimator
that adding Bf to y, with arbitrary f 2 R
p
, results in a
shift of l
T
b
f. After all, we would not like the integer part
of the estimator to become contaminated by such an
addition to y. Estimators of h that fulfil these two con-
ditions will be called integer equi variant (IE). Hence,
they are defined as follows.
Definition 2: integer equivariant (IE) estimators.The
estimator
^
hh
IE
¼ f
h
ðyÞ , with f
h
: R
m
7!R, is said to be an
IE estimator of h ¼ l
T
a
a þ l
T
b
b if
f
h
ðy þ AzÞ¼f
h
ðyÞþl
T
a
z; 8y 2 R
m
; z 2 Z
n
f
h
ðy þ BfÞ¼f
h
ðyÞþl
T
b
f; 8y 2 R
m
; f 2 R
p
ð10Þ
It is not difficult to verify that the integer estimators of
the previous section are IE. Simply check that the above
two conditions are indeed fulfilled by the estimator
hh ¼ l
T
a
aa þ l
T
b
bb. The converse, however, is not necessarily
true. The class of IE estimators is therefore a larger class.
We will now show that the class of IE estimators is
also larger than the class of linear unbiased estimators.
Let f
T
h
y, for some f
h
2 R
m
, be the linear estimator of
h ¼ l
T
a
a þ l
T
b
b. For it to be unbiased we require, using
Efyg¼Aa þ Bb, that f
T
h
Aa þ f
T
h
Bb ¼ l
T
a
a þ l
T
b
b,
8a 2 R
n
; b 2 R
p
holds true, or that both l
a
¼ A
T
f
h
and
l
b
¼ B
T
f
h
hold true. But this is equivalent to stating that
f
T
h
ðy þ AaÞ¼f
T
h
y þ l
T
a
a; 8y 2 R
m
; a 2 R
n
f
T
h
ðy þ BbÞ¼f
T
h
y þ l
T
b
b; 8y 2 R
m
; b 2 R
p
ð11Þ
Comparing this result with Eq. (10) shows that the
condition of linear unbiasedness is more restrictive than
the condition of integer equivariance. Hence, the class of
linear unbiased estimators is a subset of the class of IE
estimators. This result also automatically implies that IE
estimators exist which are unbiased. Thus, if we denote
the class of IE estimators as IE, the class of unbiased
estimators as U, the class of unbiased IE estimators as
IEU, the class of unbiased integer estimators as IU, and
the class of linear unbiased estimators as LU, we may
summarize their relationships as: IEU ¼ IE \ U 6¼;,
LU IEU and IU IEU (see Fig. 1).
3.2 Representation of IE estimators
In order to obtain a better understanding of how IE
estimators operate, it would be useful to have a
representation that reveals their structure. One such
representation is given in the following lemma.
Lemma 1: IE-representation. Let
^
hh
IE
¼ f
h
ðyÞ be the IE-
estimator of h ¼ l
T
a
a þ l
T
b
b, let y ¼ Aa þ Bb þ Cc, with
the m ðm n pÞ matrix C chosen such that ðA; B; CÞ is
invertible, and let g
h
ða; b; cÞ¼f
h
ðAa þ Bb þ CcÞ. Then
functions h
h
: R
n
R
mnp
7!R exist such that
405
g
h
ða; b; cÞ¼l
T
a
a þ l
T
b
b þ h
h
ða; cÞð12Þ
with h
h
ða þ z; cÞ¼h
h
ða; cÞ for all z 2 Z
n
.
Proof of Lemma 1 is given in the appendix.
This representation turns out to be useful in some of
the later proofs. Also note that Lemma 1 now easily
allows us to design our own IE estimator. When devising
our own IE estimator, there are essentially two types of
degrees of freedom involved: the choice of the matrix C
and the choice of the function h
h
.
The following are some examples of IE estimators
obtained for specific choices of C and h
h
.
Example 1. For arbitrary C and h
h
¼ 0 we obtain
^
hh
IE
¼ l
T
a
a þ l
T
b
b
Note that this is a linear unbiased estimator of h for any
choice of C. Hence, matrix C governs the choice of these
linear unbiased estimators.
Example 2. For h
h
¼ 0andC chosen such that
C
T
Q
1
y
ðA; BÞ¼0 we obtain the LS estimator
^
hh
IE
¼ l
T
a
^
aa þ l
T
b
^
bb
Example 3. For h
h
ða; cÞ¼ðl
T
a
þ l
T
b
Q
^
bb
^
aa
Q
1
^
aa
Þða SðaÞÞ
and C chosen such that C
T
Q
1
y
ðA; BÞ¼0 we obtain the
estimator
^
hh
IE
¼ l
T
a
aa þ l
T
b
bb
Example 4. For h
h
ða; cÞ¼ðl
T
a
þ l
T
b
Q
^
bb^aa
Q
1
^aa
Þða SðaÞÞ,
SðaÞ¼arg min
z2Z
n
ða zÞ
T
Q
1
^
aa
ða zÞ and C chosen such
that C
T
Q
1
y
ðA; BÞ¼0 we obtain the integer LS estimator
^
hh
IE
¼ l
T
a
aa
ILS
þ l
T
b
bb
ILS
4 Best integer equivariant estimation
4.1 The BIE estimator
Having defined the class of IE estimators we will now
look for an IE estimator which is ‘best’ in a certain
sense. We will denote our best integer equivariant (BIE)
estimator of h as
^
hh
BIE
and use the MSE as our criterion
of ‘best’. The BIE estimator will therefore be defined as
^
hh
BIE
¼ arg min
f
h
2IE
Efðf
h
ðyÞhÞ
2
gð13Þ
in which IE stands for the class of IE estimators. The
minimization is thus taken over all IE functions that
satisfy the conditions of Definition 2.
The reason for choosing the MSE criterion is two-
fold. First, it is a well-known probabilistic criterion for
measuring the closeness of an estimator to its target
value, in our case h. Second, the MSE criterion is also
often used as measure for the quality of the ‘float’
solution itself. The following theorem gives the solution
to the above minimization prob lem of Eq. (13).
Theorem 1: best integer equivariant estimation. Let
y 2 R
m
have mean Efyg¼Aa þ Bb and pdf p
y
ðyÞ, and
let
^
hh
BIE
be the BIE estimator of h ¼ l
T
a
a þ l
T
b
b. Then
^
hh
BIE
¼
P
z2Z
n
R
R
p
ðl
T
a
zþl
T
b
bÞp
y
ðy þAða zÞþBðb bÞÞdb
P
z2Z
n
R
R
p
p
y
ðy þAðazÞþBðbbÞÞdb
ð14Þ
Proof of Theorem 1 is given in the appendix.
Note that the BIE estimator can also be written as
^
hh
BIE
¼ l
T
a
^
aa
BIE
þ l
T
b
^
bb
BIE
ð15Þ
with
^
aa
BIE
¼
X
z2Z
n
zw
z
ðyÞ;
X
z2Z
n
w
z
ðyÞ¼1
^
bb
BIE
¼
Z
R
p
bw
b
ðyÞdb;
Z
R
p
w
b
ðyÞdb ¼ 1
in which the weighting functions w
z
ðyÞ and w
b
ðyÞ are
defined by Eq. (14). This shows that the BIE estimator
of the integer parameter vector a is also a weighted
sum of all integer vectors in Z
n
, just like
aa of Eq. (6) is.
In the present case, however, the weights are not
binary. They vary between 0 and 1, and their values are
determined by y and its PDF. As a consequence the
estimator
^
aa
BIE
will in general be real valued, instead of
integer valued.
The above theo rem holds true for any PDF the vector
of observables y might have. This is therefore a very
general result indeed. A closer look at Eq. (14) reveals
however, that we need a and b, and therefore h, in order
to compute
^
hh
BIE
. The dependence on a and b is present
in the numerator of Eq. (14) and not in its denominator.
The summation over all integer vectors in Z
n
and the
integration over R
p
makes the dependence on a and b
disappear in the denominator. If the dependence of
^
hh
BIE
on h persisted we would not be able to compute the BIE
estimator. Note, however, that this dependence disap-
pears in the case that the PDF of y has the structure
p
y
ðyÞ¼f ðy Aa BbÞ; this property is fortunately still
true for a large class of PDFs.
Fig. 1. The set relationships between the different classes of estima-
tors: integer equivariant estimators IE, unbiased estimators U,
unbiased integer equivariant estimators IEU, unbiased integer
estimators IU, and linear unbiased estimators LU
406
