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GEODESY AND CARTOGRAPHY
Vol. 66, No 1, 2017,
© Polish Academy of Sciences
DOI: 10.1515/geocart20170001
MP estimation applied to platykurtic sets
of geodetic observations
Zbigniew Wiśniewski
University of Warmia and Mazury
Institute of Geodesy
1 Oczapowskiego St., 10957 Olsztyn, Poland
email: zbyszekw@uwm.edu.pl
Received: 3 November 2016 / Accepted: 23 November 2016
Abstract: MP estimation is a method which concerns estimating of the location parameters
when the probabilistic models of observations differ from the normal distributions in the
kurtosis or asymmetry. The system of Pearson’s distributions is the probabilistic basis for
the method. So far, such a method was applied and analyzed mostly for leptokurtic or
mesokurtic distributions (Pearson’s distributions of types IV or VII), which predominate
practical cases. The analyses of geodetic or astronomical observations show that we may
also deal with sets which have moderate asymmetry or small negative excess kurtosis.
Asymmetry might result from the inﬂ uence of many small systematic errors, which were
not eliminated during preprocessing of data. The excess kurtosis can be related with
bigger or smaller (in relations to the Hagen hypothesis) frequency of occurrence of the
elementary errors which are close to zero. Considering that fact, this paper focuses on
the estimation with application of the Pearson platykurtic distributions of types I or II.
The paper presents the solution of the corresponding optimization problem and its basic
properties.
Although platykurtic distributions are rare in practice, it was an interesting issue to
ﬁ nd out what results can be provided by MP estimation in the case of such observation
distributions. The numerical tests which are presented in the paper are rather limited;
however, they allow us to draw some general conclusions.
Keywords: M and MP estimation, platykurtic probabilistic models, Pearson’s
distributions
1. Introduction
Considering the classical theory of measurement errors, we usually assume that the
Gauss distributions (the normal distributions) are their probabilistic models. The
family of such distributions corresponds with the hypothesis of the elementary errors
given by Hagen and Bessel (e.g. Fischer, 2011). However, the analyses show that the
empirical distributions of errors of geodetic, geophysical or astronomical observations
might often differ from the normal ones. The basic anomalies in this context concern
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Pearson’s squared skewness
E
PP
and/or the kurtosis
E
PP
(μk – the
kth central moment). Besides the coefﬁ cient β1, one can also apply the skewness
VJQ
J
PP P E
, which allows us to determine the sign of the asymmetry
(positive or negative). Note that for the normal distributions β1 = 0 and β2 = 3. Due to
such a value of the kurtosis, anomalies of other distributions in this context are often
described by the excess kurtosis γ2 = β2 – 3 (e.g. Dorić et.al., 2009).
Asymmetry might result from the inﬂ uence of many small systematic errors, which
were not eliminated during preprocessing of data. Then, the axiom which concerns
the same number of positive and negative errors, and which is given in the classical
theory of measurement errors, is not met (e.g. Pearson, 1920; Friori and Zenga, 2009).
Kukuča (1967) and Dzhun’ (2012) indicated such a reason of asymmetry of the error
distributions in the case of geodetic or astronomical observations. If the systematic
errors are carefully eliminated, then the skewness usually achieves the small values.
For example, in the case of the astrometric observations within the project MERIT,
β1 = 0.0048 (Dzhun’, 2012); for the phase measurements from the SAPOS®, GNSS
observations, β1 = 0.0121 (Luo et al., 2011). Similar values for GPS observations
were also obtained by Tiberius and Borre (2000).
The excess kurtosis can be related with bigger or smaller (in relations to the
Hagen hypothesis) frequency of occurrence of the elementary errors which are
close to zero. The surfeit of such errors is the origin of leptokurtic distributions
(β2 > 3), and the deﬁ ciency – platykurtic distributions (β2 < 3). Note that distributions
are mesokurtic when β2 = 3. Romanowski and Green (1983) noted that observation
errors have usually symmetric mesokurtic or leptokurtic distributions, which justiﬁ ed
application of the modiﬁ ed normal distributions. Except for small asymmetry, such
a note is consistent with other empirical analyses. For example, Dzhun’ (1992, 2012)
showed that the errors of the astronomic observations have usually the kurtosis
β2 = 3.8 (however, β2 = 4.858 was obtained during the project MERIT). Similar values
of the kurtosis were obtained by Wassef (1959) and (Kukuča 1967) in the precise
leveling. Considering contemporary observations, we should expect a wider range
of the kurtosis values. For example, in the case of the observations from Satellite
Laser Ranging, the kurtosis ranging β2 = 2.69 ÷ 9.46 (Hu et al., 2001), and for GPS
observations β2 = 2.79 ÷ 3.29 ( Luo et al., 2011).
One should note that the asymmetric coefﬁ cient and the kurtosis are usually
estimated based on the sample moments (e.g. Dorić et al., 2009). Another possible
approach is to compute all necessary moments during the process of adjustment by
the least squares method (Wiśniewski, 1996; Kasietczuk, 1997). The statistic tests
are the basis for determining whether the anomalies obtained are relevant, and the
empirical distribution cannot still be described by the normal distribution. Here, the
JarqueBera test or the D’agostino test (D’agostino et.al, 1990; Kayikçi and Sopaci,
2015) can be applied. Note, that the kurtosis estimate might be affected with gross
errors (Kukuča, 1967). Thus, before application of the signiﬁ cance testing for β1 and
β2, it is advisable to detect outliers by applying any of the methods presented in
(Barda,1968; Lehmann, 2012).
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If the asymmetry and/or kurtosis are signiﬁ cant, then one should decide which
theoretical distribution is adequate for the measurement errors. In the case of symmetric
leptokurtic distributions, one can apply several different distributions including
the modiﬁ ed normal distributions given by Romanowski (1964) or the generalized
normal distribution with the shape parameter which is steered by the excess kurtosis.
Lehmann (2015) proposed to apply an information criterion, for example the Akaike
Information Criterion, when a suitable probabilistic model is selected (besides the
statistical hypothesis and test). The author presented pros and cons of such a method,
considering the generalized normal distribution and its special cases.
In the case of wide range of asymmetry coefﬁ cient and/or kurtosis, the choice
of a particular probabilistic model might be a complicated issue. Then the Pearson
Distribution system (PDsystem), which was proposed by Pearson (1920), seems to
be a convenient solution. The distributions that belong to such a system are directly
steered by the coefﬁ cients β1 and β2, and are very stable when approximating
empirical distributions. Xi et al. (2012) showed that similar stability concerns also the
saddlepoint approximation, the maximum entropy principle or the Johnson system.
However, PDsystem gives better results for small asymmetry and moderate values
of the kurtosis. The general properties of the Pearson distributions are discussed by
Elderton (1953) or Friori and Zenga (2009). The several selected distributions of that
system were applied in astronomy (Dzhun’, 1992, 2012) and in geodesy (Wiśniewski,
1987, 2014).
One of the main issues of adjustment process is to estimate the parameters of
a functional model of observations. Usually, the least squares method (LSmethod) is
applied in such a case. However, if we know the probability density functions (PDF)
of the measurement errors, then application of the maximum likelihood method
(MLmethod) seems more justiﬁ ed, e.g. Serﬂ ing (1980). Considering more general
assumptions concerning the probabilistic models, one can also apply Mestimation
which is based on a particular inﬂ uence function or a weight function (Huber,1964,
1981). Wiśniewski (2009, 2010) proposed another generalization of Mestimation,
namely Msplit estimation, where the main assumption is that there are several
competitive functional models which can be related to the particular observation set
(see also Duchnowski and Wiśniewski 2011, 2014, 2016).
Disturbances in estimation process that are caused by anomalies of empirical
distributions, were discussed in general, for example, in Mooijaart (1985) or
Mukhopadhyay (2005), and in the case of geodetic networks and LSmethod in
Gleinsvik (1971) and Wiśniewski (1985). Wiśniewski (1987) proposed to include
such anomalies into the computation by applying and selected Pearson’s distributions.
Such a method requires the knowledge of the particular PDF, which is the basis for
a newly formulated optimization problem. Dzun’ (2011) showed that the adjustment
that is based on MLmethod and Pearson’s distributions can be performed in a simpler
way. The idea behind such an approach is the application of a weight function and the
knowledge that PDFs of Pearson’s distributions are solutions of a particular differential
equation. The differential expression included in such an equation is proportional to
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the inﬂ uence function, which is very suitable here. Note that the inﬂ uence function is
based on distribution functions for whole PDsystem. Considering such assumptions
and the main idea proposed by Dzun’a (2011), Wiśniewski (2014) brought and
analyzed a new solution called MP estimation. The paper in question focused on such
variants of the method which are referred to mesokurtic or leptokurtic distributions.
Actually, such distributions predominate in astronomical and geodetic observations;
however, they do not cover all the possible cases (e.g. Hu et al., 2001; Luo et al.,
2011). Thus, there is a need for consideration MP estimation for distributions which
excess kurtosis is negative. We should realize that the application of the differential
equation ΩPD means that we do not choose any particular probabilistic model in fact
(or any particular PDF), which is very important in such a context. We do not refer to
the general properties of PDF, but we applied the values of the excess kurtosis and the
asymmetry coefﬁ cient. Thus, MP estimation is steered only by those two coefﬁ cients
and by the standard deviation.
The paper is organized in the following way: Section 2 recalls the general
assumptions of Mestimation based on the application of the inﬂ uence and weight
functions; Section 3 presents application of the Pearson system of distributions in
MP estimation. The special attention is paid to the Pearson distributions of types
I and II, which are platykurtic. Finally, Section 4 presents results of numerical tests.
Although these tests are elementary, they allow some general conclusions to be drawn
(Section 5).
2. MEstimation
The following functional model is usually assumed in theory and practice
y = AX + v (1)
where y Rn is an observation vector, A Rn,r is a known matrix of coefﬁ cients
(rank (A) = r), X Rr is a vector of unknown parameters and v Rn is a vector
of random errors. The elements vi of the vector v are assumed to be independent
and their distributions Pθi are indexed with the parameter θ Θ (Θ is a parameter
space). The distributions Pθi which belong to the family
^ `
L
L
3
T
T4
P
are regarded
as probabilistic models of observation errors. In addition, we assume that the
distribution PX is a probabilistic model of the observation yi = aiX + vi (ai – ith row of
the matrix A). Note that such a distribution is indexed with the vector of parameters
X (Θ = Rr).
Consider the classical variant of Mestimation, then one should solve the following
optimization problem (Huber,1981; Hampel et al., 1986):
PLQ
QQ
0LL
LL
\Y
M
UU
¦¦
;;
(2)
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MP estimation applied to platykurtic sets of geodetic observations
where ρ(yi; X) = ρ(yi – aiX) = ρ(vi). This is a generalization of MLmethod, which
optimization problem can be written as
>OQ @ >OQ@ PLQ
QQ
0/ L L
LL
I\ IY
M
¦¦
;;
(3)
where f(yi; X) = f(vi) is PDF (or it is proportional to PDF). Thus, a particular family
of distributions indexed with the parameter vector X should be assumed. In the case
of Mestimation, the functions ρ(vi) are arbitrary; however, considering the following
relation
OQ
LL
YIY
U
H[S> @
LL
I
YF Y
U
U
(4)
(c > 0 is a normalization parameter) such functions can also be referred to certain
distribution families. For example, the Huber method (Huber1964, 1981) assumes
that the probabilistic model is deﬁ ned by the family of the generalized normal
distributions with twosegment PDF (Lehmann 2015).
Considering the function φM(X), one can write the following respective
gradient
7
77 7 7
QQ Q
LL L
0
LL
LL L
L
GY Y Y
YY
GY
U
MU\
ww
ww
ww w w
¦¦ ¦
;
J; $ȥY
;; ; ;
(5)
where ψ(v) = [ψ(vi), ..., ψ(vn)]T. Thus, Mestimates of the vector X fulﬁ lls the equation
(Huber,1981; Hampel et al., 1986)
7 $ȥY
(6)
The functions ψ(vi) are proportional to the inﬂ uence functions IF(vi, FX) which
are based on the distribution functions F(yi; X) = FX(vi) F. Here, F is a family
of distribution functions namely F = {FX(vi) : X Rr}, which corresponds with
the family of distributions of the observation errors P = {Pθi : θi Θ} (Hampel,
1974; Serﬂ ing, 1980; Hampel et al., 1986). The functions ψ(vi) are also often called
the inﬂ uence functions. If the components ρ(vi) of the objective function in the
optimization problem of Eq. (2) are known, then the inﬂ uence functions ψ(vi) can be
written in the following way
LLL L
LLLL
LLL
L
GY GY GY GY
YZYYZY
GY GY GY
GY
UU
\
(7)
where
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L
L
L
GY Y
ZY Y
GY
U\
(8)
is a weight function (Huber, 1981; Yang, 1997). If additionally ρ(vi) = ln f(vi), then
one can also write that
OQ
LL L
L
LL LL
GY G Y GIY
YGY GY I Y GY
U
\
(9)
Let us introduce the diagonal weight matrix w = diag(w(v1),..., w(vn)), then the
vector of the inﬂ uence function can be written as ψ(v) = 2w(v)v. This leads to another
form of Eq. (6), namely
77
J; $ZYY $ZY \ $;
(10)
Mestimate which is its iterative solution has the following form
77
ÖÖÖ
>@
; $ZY$ $ZY\
(11)
For
Ö
ZY 3
, where
GLDJ
Q
VV
v3
, one can obtained LSestimate of the
parameter vector X. The estimate
77
Ö
/6
;
$3$ $3\
is a very convenient and
often applied starting point within the iterative process which leads to the solution
of Eq. (11).
Wiśniewski (2014) proposed the name MP estimates for all the solutions of Eq.
(7) which are based on the inﬂ uence functions ψ(vi) IF(vi, FX) and the explicit
families of distributions P = {Pθi : θi Θ}. In such a context, most of the popular
Mestimates are also MP estimates, including the Huber estimates (the generalized
normal distributions) or LSestimates (the Gauss distributions).
3. MP Estimation with PDSYSTEM
Wiśniewski (2014) proposed and analyzed the case of MP estimation, in which
PPD = {Pθ : θ Θ} is a family of Pearson’s distributions (PDsystem). The origin of
PDsystem is the following differential equation Ω* (Pearson, 1920; Elderton, 1953;
Dzhun’, 2011)
FFYF
GI Y
YIY GY FFYFY
V
:VV
(12)
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MP estimation applied to platykurtic sets of geodetic observations
where: c0 = 4β2 – 3β1, c1 = γ1(β2 + 3), c2 = 2β2 – 3β1 – 6,
VJQ
J
PE
. However,
there might be some problems in direct application of that equation in MP estimation.
Note that in such a case, the expected value E(v) lays at the origin of the coordinate
system. If we assume that the estimate
Ö
;
should minimize the amount of information
of a particular observation set, then the mode M0 should lay at the origin of the
coordinate system. In the case of asymmetric distribution, the mode M0 does not
coincide with the expected value E(v). Considering such a requirement, Wiśniewski
(2014) proposed to modify the differential equation of PDsystem in the following
way
FFY
GI Y
YY
IY GY FFYVFYV
:\
VV
(13)
where s = M0 – E(v) = σc1/(c0 + 3c2). Taking in account the properties of the inﬂ uence
function ψ(v) = –Ω(v) in the context of MP estimation, it is worth considering two
versions of that function, namely
OHS
SOW
YF IRU
YYF IRU
\JEEJ
\\JEEJ
tt
®
¯
(14)
wherein, if β2 = 3 then ψlep(v) = ψmez(v), and ψmez(v) is the inﬂ uence function given for
mesokurtic distributions (lep – leptokurtic distributions, plt – platykurtic distributions).
For
NF FF
(the Pearson distributions of types IV and VII), the inﬂ uence
function ψlep(v) and the corresponding weight function
OHS
OHS
Y
ZY Y
\
(15)
have unlimited range. MP estimation with such a weight function was discussed in
detail in (Wiśniewski, 2014). Such a property, which is advisable from the practical
point of view, does not apply to the inﬂ uence functions in the case of platikurtic
distributions. For
NF FF !
(the Pearson distributions of types I and II), the
inﬂ uence function ψplt(v) and the corresponding weight function
SOW
SOW
Y
ZY Y
\
(16)
have limited domain δv = (a1,a2), where
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P]
DPP
V
P
DD
P
FFF
]
F
PP

PP

FF
PF
FF
]F
J

(17)
For β1 = 0, it holds that –a1 = a2, where
D
EE
D
EE
.
If β2 → 3, then a1 → –∞ and a2 → +∞.
Some variants of the inﬂ uence functions and the corresponding weight functions,
which are obtained for several values of the kurtosis and for the asymmetry coefﬁ cient
equal to zero, are presented in Figure 1.
Fig. 1. Inﬂ uence and weight functions in the case of symmetric probabilistic models
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MP estimation applied to platykurtic sets of geodetic observations
The weight functions wlep(v) have unlimited range and are bellshaped (symmetric for
β1 = 0 or asymmetric for β1 > 0). What is more, supvwlep(v) = w(M0). Considering the
general classiﬁ cation of Mestimates (Kadaj, 1988; Wiśniewski, 2014), one can say
that the following MP estimate
77
ÖÖÖ
>@
OHS OHS OHS
; $ZY$$ZY\
(18)
satisﬁ es the condition K (it is a robust estimate). If β1 = 0, β2 = 3, then c0 = 2, c1 = 0,
c2 = 0. Thus, we obtain ψlep(v) = ψmez(v) = ψLS(v) = v/σ2 and wLS(v) = ψLS(v1)/ v1= 1/σ2,
which are the inﬂ uence function and the weigh function of LSmethod, respectively
(and the method itself satisﬁ es the condition K0 – neutral estimation).
If β2 < 3, then the weight functions wPD(v) = wplt(v) are Ushaped within the
interval δv= (a1, a2). Therefore, they have two upper bounds within the interval
YDHDH TT
'
¢ ² ¢ ²
, e > 0, namely
VXS
SOW SOW Z
TY0
ZY ZT
O
d
VXS
SOW SOW Z
0YT
ZY ZT
O
d
(19)
wherein, if e → 0, then
Z
O
of
DQG
Z
O
of
. Since
PLQ
ZZ
Z0
OO
, then
MP estimates
Ö
;
plt, which solves the equation ATwplt(v)v = 0, satisfy the condition
K+ within the interval δv (weak estimation). As a consequence, especially when the
number of observations is low, the following iterative process
7 7
>@
MM M
SOW SOW
; $ZY$$ZY\
MM
SOW
Y\$;
(20)
might be divergent. This results from the fact that the vector X j+1 is computed
by applying the weight matrix, wplt(v j) , which depends on the residuals from the
previous iterative step. Because the weight function is convex, the estimates X j
and X j+1 move away from each other and tend to the respective boundary points of
the interval δv. This would be an interesting property of the method; however, this
also raises a problem. While the estimates approach the boundary points a1 or a2,
respectively, some iterative residuals might be out of the interval δv = (a1, a2). The
weight functions do not exist outside such interval (or they achieve unrealistic values),
thus the iterative process cannot stabilize (without any special intervention, like for
example, introduction of external artiﬁ cial weigh functions). One should note that the
cross weighting leads to interesting results in the case of a weak estimation, but only
if the interval δv is inﬁ nite. A good example of such an approach is Msplit estimation
which is based on a split of Mestimate with the application of the weight function
w(v) = v2 (Wiśniewski, 2009, 2010; Cellmer, 2014; Wiśniewski and Zienkiewicz,
2016).
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It is noteworthy that for the growing kurtosis, the weight function ﬂ attens within
the certain interval Δv δv. Considering a particular interval
YNN
'VV
¢ ²
, such
ﬂ attening might be analyzed by application of the following differences (for k > 0)
/SOW SOW
UZ N Z0
V
5SOW SOW
UZN Z0
V
(21)
Table 1 presents the values of rL and rR for several values of β1 and β2 (the skewness
is positive) and under the assumption that k = 2.5 and σ = 1.
Table 1. Differences rL and rR in relation to asymmetric coefﬁ cient β1 and kurtosis β2
β2 = 2.9999999 β2 = 2.99 β2 = 2.90 β2 = 2.70 β2 = 2.30
β1 = 0.00
rL0.000 0.010 0.114 0.443 10.568
rR0.000 0.010 0.114 0.443 10.568
β1 = 0.01
rL0.212 0.027 0.131 0.474 20.253
rR0.187 0.002 0.099 0.415 7.102
β1 = 0.04
rL 0.226 0.051 0.149 0.507 201.285
rR0.176 0.002 0.086 0.389 5.519
Kutterer (1999) considered the problem of how LSestimation is inﬂ uenced by
disturbances of the weighs (herein, such disturbances correspond with the values of
the differences rL and rR which decrease with the excess kurtosis tending to zero
from the lefthand side). Generally, such inﬂ uence should be analyzed separately
from the theoretical point of view. However, for moderate negative values of the
excess kurtosis and small values of the skewness, one can expect that
Ö
;
plt =
Ö
;
LS.
If the anomalies of the empirical distribution are bigger, then such equality might
not be true. In such a case, one can get satisfactory results if the observations are
concentrated around the mode in a sufﬁ cient way or, in other words, if the errors of
the observations which are grouped around the mode have the decisive inﬂ uence on
value of the expression
Q
LL
L
ZY \
¦
. Although the weights of those observations are
the smallest, the great number of the errors which are close to the mode might make
the iterative process of Eq. (20) to converge. The chance for a satisfactory solution
increases with the growing number of observations.
If the number of observations is small, then the empirical distribution might have
some local modes (apart from the global one). Then, in the ﬁ nal steps of the iterative
procedure of Eq. (20), the process might stabilize and generate sequence of some
repeated values. The estimate that we are interested in, and which zeroes the gradient
g(X) = –AT wplt(v)v might be among such values. However, such estimation way is
not convincing from the theoretical as well as practical point of view. Generating of
the sequence of some repeated values usually results from unsuitable starting point
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MP estimation applied to platykurtic sets of geodetic observations
or relatively big iterative increases (especially in the very ﬁ rst iterative steps). If the
skewness is moderate, the estimate
Ö
;
LS is usually close to the mode. Thus, if one
assumes that X0 =
Ö
;
LS, then it might happen that after the ﬁ rst iterative step the
process “skips” the global mode and tends to one of the local ones. To avoid such
a situation, one can apply the Newton method with the correction which forces the
reduction of the iterative increases between subsequent iterative steps. Hessian of the
objective function φM (X), which is necessary in such an approach, can be written as
follows
7
7
0
SOW
J
M
ww
w
ww
;;
+; $Z Y$
;
;;
(22)
Thus, the iterative process can be written in the following way (j = 1,..., m)
77
> @ > @
MMM M M
M
SOW SOW
MM M
MM
G
G
W
; +; J; $ZY$$ZYY
;; ;
Y\$;
(23)
where τ < 1 is a reduction coefﬁ cient of the iterative increase. One should assume
the value of τ so that the iterative process of Eq. (23) is convergent and ends up at the
point
Ö
;
= Xj=m,
Ö
Ö Y\$;
, for which
7
ÖÖÖ
SOW
J; $Z YY
.
4. Numerical examples
Let us assume the following functional model of the observations vi = yi – X,
i = 1,...,n, wherein X is an estimated parameter. From the geodetic point of view, such
a mode might relate, for example, to a leveling network with one unknown point
W and some ﬁ xed points Pj, j = 1,...,k (see, Figure 2). Let hj be a height difference
between the points Pj and W, and let each hj be measured sj times, then
N
M
M
QV
¦
.
In the context of the model in question, measurements of hj are observations yj, and
the height of the point W is the parameter X.
The observations are simulated by applying the Gaussian generator randn(n,1)
of the system MatLab. For small numbers of the observations n, the sample
distributions often differ from the given normal distribution in the skewness or the
kurtosis (Wiśniewski 2014). Such sets usually have some local aggregations of the
observations. For bigger values of n, observation sets generated by randn(n,1) have
the assumed theoretical properties. For that reason, to obtain sets with negative excess
kurtosis or signiﬁ cant skewness, one can combine normal distributions with different
expected values (but with the control of the assumed standard deviation of the whole
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set). Sets of the observations yi, i = 1,...,n, will be denoted as ω, where l is the number
of a set.
Fig. 2. Simulated, elementary leveling network
Consider some sets ω1 which were generated under the assumption that X = 0. Let the
empirical moments be computed for each of such sets. Hence σ, γ1 and β2 can also be
computed. Thus, one can compute the following statistic

%Q
EE
ªº
«»
¬¼
(24)
which is the basis for the JarqueBera test (Bera and Jarque, 1980). Such a statistic
is distributed according to the
D
F
distribution with two degrees of freedom if only
the null hypothesis
a
+
\1
V
is true (N  normal distribution, α – signiﬁ cance
level). If JB <
D
F
, then the null hypothesis is not rejected (the observations are from
a normal distribution). Table 2 presents some critical values of the variable
D
F
for
given signiﬁ cance levels.
Table 2. Critical vales of
D
F
in the JarqueBera test
α0.025 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
D
F
7.378 5.991 4.605 3.219 2.408 1.833 1.386 1.022 0.713 0.446 0.211
The basic analysis of the estimation for platikurtic distributions is based on the
observation sets for which n = 300. The empirical distributions of two example sets
are presented in Figure 3. The results obtained by applying the iterative process
of Eq. (20) are presented in Table 3 (all the iterative processes were convergent).
Table 3 presents the empirical distribution parameters, namely σ, γ1, β2, the boundary
points of the interval δv = (a1, a2), and ﬁ nally MP and LS estimates of the parameter
X. The last column shows the signiﬁ cant levels α*, for which the null hypothesis
about normality of the distribution should be rejected.
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Fig. 3. Example empirical distributions for n = 300
Table 3. MP and LS estimates of X = 0 for platikurtic distributions (n = 300)
Sets
Empirical parameters Acceptable interval Estimators
JB α*
σγ
1β2a1a2MPLS
ω1 1.130 0.395 2.985 8.994 3.097 0.104 0.369 7.818 0.025
ω21.094 0.384 2.818 6.088 2.476 0.036 0.291 7.808 0.025
ω31.206 0.382 2.893 7.785 3.023 0.088 0.397 7.452 0.025
ω41.068 0.348 2.964 3.285 8.811 0.004 0.231 6.071 0.05
ω51.203 0.298 2.689 2.689 5.699 0.006 0.289 5.664 0.10
ω61.178 0.077 2.405 3.042 3.616 0.192 0.293 4.726 0.10
ω71.022 0.249 2.768 3.037 5.620 0.003 0.174 3.774 0.20
ω81.060 0.041 2.499 3.042 3.616 0.082 0.120 3.208 0.30
ω91.042 0.132 2.593 3.082 4.174 0.126 0.224 2.994 0.30
ω10 0.959 0.044 2.537 3.011 3.323 0.009 0.044 2.774 0.30
ω11 1.039 0.158 2.709 5.096 3.455 0.024 0.129 2.293 0.40
ω12 0.947 0.006 2.644 3.624 3.680 0.042 0.050 1.581 0.50
ω13 1.024 0.024 2.653 4.122 3.883 0.042 0.021 1.534 0.50
ω14 1.006 0.065 2.961 14.049 9.598 0.046 0.080 0.228 0.90
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Similar results were also obtained for other tests when n = 300. The difference
between MP and LS estimates of the parameter X usually decreases with decreasing
values of the statistic JB. However, it is not a general rule since the statistic JB
can achieve the similar values for different skewness and excess kurtosis. The
asymmetry (skewness) has the biggest inﬂ uence on the difference between the
estimates in question. If it is small, the difference is also not so signiﬁ cant (see, the
sets ω11, ω12, ω13); if the skewness is very close to zero, then usually the MP and
LS estimates are equal to each other. On the contrary, if the negative excess kurtosis
is very small and the skewness is large (deviation from the normal distribution is
very signiﬁ cant), then the differences between the estimates are distinct (see, the sets
from ω1 to ω4).
For the big observation sets, the results of MP estimation obtained for platikurtic
distributions are similar to those for leptokurtic distributions (if only the observation
sets are free of outlying observations). If the number of observations is smaller,
then some local modes might occur. In the case of leptokurtic distributions, the
weight functions are concave; hence, such local modes do not result in any serious
optimization problems. On the other hand, some problems with ﬁ nding the minimum
of the objective function might happen for platykurtic distributions, for which the
weight functions are convex (this was mentioned in the previous section). Table 4
presents the estimates of the parameter X, for the observation sets for which n = 32.
The empirical distributions of two of them are presented in Figure 4. Note that for
small observation sets and a reasonable signiﬁ cance level, the values of the statistics
JB do not suggest that the hypothesis about the normality of observation distributions
should be rejected (even if excess kurtosis or asymmetry is signiﬁ cant). However, this
does not mean that MP estimation cannot provide better results than the conventional
LS estimation.
Fig. 4. Empirical distributions for n = 32
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Table 4. MP and LS estimates of parameter X = 0 for platykurtic distributions (n = 32)
Sets
Empirical parameters Estimates
JB
Comments about estimation
process
of Eqs. (20 or 23)
σγ
1β2MPLS
ω15 0.915 0.053 2.010 0.065 0.116 1.321
Process (20) divergent
Process (23) convergent for
τ = 0.1
ω16 1.034 0.191 2.202 0.042 0.109 1.044
Process (20) divergent
Process (23) convergent for
τ = 0.2
ω17 0.842 0.297 2.471 0.037 0.249 0.842 Process (20) convergent
For the sets ω15, ω16, the iterative processes of Eq. (20), which solve the equation
ATwplt(v)v = 0, were divergent. Thus in such cases, the Newton method with
a reduction coefﬁ cient τ was also applied. The iterative process for the observation set
ω16 is presented in Table 5 (τ = 0.2). As for the observation set ω15, the gradient was
zeroed (with the assumed tolerance) for τ = 0.1, which make the iterative process much
longer. In the case of the set ω17, the iterative process of Eq. (20) was convergent and
ended up after 15 iterative steps (similar number of the iterative steps was obtained
when the gradient method for τ = 1 was applied).
Table 5. Iterative process (the Newton method) for set ω16
Steps jg(X j) H(X j) dX j X j+1
0X 0 = X LS = 0.1092
1 7.126 19.401 0.0735 0.0356
2 0.480 15.693 0.0061 0.0418
3 0.005 15.909 6.6 10–5 0.0418
4 6.2 10–5 15.910 7.9 10–5 0.0418
5 7.5 10–7 15.910 9.4 10–9 0.0418
5. Conclusions
Although platykurtic distributions are rare in practice, it was an interesting issue to
ﬁ nd out what results can be provided by MP estimation in the case of such observation
distributions. The numerical tests which are presented in the paper are rather limited;
however, they allow us to draw some general conclusions.
The weight function in MP estimation is convex if one applies the platykurtic
Pearson distributions of types I or II. For big observation sets, which excess kurtosis
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is small and negative, such property of the weight function does not result in any
serious problems in searching for the estimate that minimizes the objective function.
In such a case, the iterative process is very similar to MP estimation with the
application of leptokurtic distributions and hence concave weight functions. Thus,
the iterative process described by Eq. (20) can be applied to solve the equation
ATwplt(v)v = 0 directly. On the other hand, some problems might occur for small
observation sets. Within such sets there might be some local modes which do not
necessarily result from the combination of various random variables. Thus, the
iterative process of Eq. (20) might be divergent or might stabilize at the sequence of
some repeated values. Note that in the case of a concave weight function (like for the
Pearson distributions of types IV or VII), the empirical local modes generally have
little chance to dominate the iterative process. Here, the signiﬁ cance of observations
which lay around the global mode is strengthened by the biggest values of the weight
function within the whole interval δv = (–∞, ∞). The situation is much different for
convex weight functions for which local modes can destabilize the iterative process.
Note that the weight functions does not exist beyond the interval δv = (a1, a2) or, like
for the Pearson distributions of types I or II, they achieve the unrealistic values hence
they cannot be regarded as real weight functions. Then, it is necessary to control
the value of the gradient for all the ﬁ nal values of the iterative process of Eq.(20).
Usually, none of such values zeroes the gradient. The iterative process may succeed
when one applies the Newton method with the correction τ < 1 which forces the
reduction of the iterative increase. However, one should assume that there is no local
“peak” between the starting point and the global mode. Such condition is usually
met if X0 =
Ö
;
LS (except some extremely unfavorable conditions when the empirical
distribution has big asymmetry and large negative excess kurtosis). If for the given
starting point the solution
Ö
;
for which g(
Ö
;
) = 0 cannot be obtained, then the iterative
process can be repeated for smaller value of the coefﬁ cient τ. It might happen that the
starting point should be changed too.
Generally speaking, MP estimation for platykurtic distributions provides similar
results like for leptokurtic ones (if only the observation set is free of outlying
observations). Both of versions of such a method, which are presented herein, can
reduce the inﬂ uence of the anomalies of empirical distributions on the ﬁ nal estimate.
Note that asymmetry of the empirical distribution disturbs the estimate in most
signiﬁ cant way. The inﬂ uence of that anomaly increases with grooving absolute value
of the excess kurtosis. In the absence of asymmetry, the mode is equal to the expected
value (for moderate negative excess kurtosis), and MP and LS estimates are also
equal to each other; however, some numerical problems with computation of MP
estimate may also occur in such a case.
The value of the test statistic JB depends most of all on the number of observations.
If such number is small, the null hypothesis (observations are normally distributed) is
rejected only for large anomalies (measured by the skewness and the excess kurtosis).
The empirical distribution can then be described by the normal one (at the reasonable
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MP estimation applied to platykurtic sets of geodetic observations
signiﬁ cance level). However, this does not mean that MP and LS estimates are always
equal to each other.
Acknowledgments
This work was supported by the University of Warmia and Mazury in Olsztyn under
Grant No: 28.610.002300.
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