Applied Mathematics, 2013, 4, 1269-1277
http://dx.doi.org/10.4236/am.2013.49171 Published Online September 2013 (http://www.scirp.org/journal/am)
Estimation of Location Parameter from
Two Biased Samples
Leonid I. Piterbarg
Department of Mathematics, University of Southern California, Los Angeles, USA
Email: piter@usc.edu
Received June 5, 2013; revised July 5, 2013; accepted July 12, 2013
Copyright © 2013 Leonid I. Piterbarg. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
We consider a problem of estimating an unknown location parameter from two biased samples. The biases and scale
parameters of the samples are not known as well. A class of non-linear estimators is suggested and studied based on the
fuzzy set ideas. The new estimators are compared to the traditional statistical estimators by analyzing the asymptotical
bias and carrying out Monte Carlo simulations.
Keywords: Biasness; Fuzzy Estimator; Location Parameter
1. Introduction
The problem is to estimate an unknown scalar parameter
from two different independent samples of size n1 and
n2
111
222
, 1,2,,,
, 1,2,,
ii
ii
1
2
x
bi n
x
bi


 
 
n
(1)
where
j
b and
j
are the bias and scale parameter of
the j-th sample respectively, j = 1, 2, and i
, i
are
zero mean independent random noises.
A novelty in our set up is that the biases
j
b, j = 1, 2
are assumed to be unknown which makes
unidenti-
fiable from the classical statistics viewpoint, e.g. [1].
Thus, traditional formulations of best estimation are not
applicable in this situation which nevertheless often
arises in applications. An important example is an as-
similation problem in physical oceanography and mete-
orology where information on a certain parameter comes
from both observations and a circulation model [2,3].
Typically such a model is biased for a variety reasons:
uncertainties in a forcing and dissipation, boundary con-
ditions, model parameters, etc. The bias in observations
is mostly due to inaccurate measurements and time/space
averaging intrinsically present in any measuring proce-
dure. That type of bias sometimes can be excluded using
a learning samples [4,5], however it is often difficult to
justify the key assumption that learning and control sam-
ples are taken from the same ensemble.
One can simply ignore the biases and apply traditional
least square or maximum likelihood methods which
would result in a biased estimate of
. As an alternative,
we suggest to use fuzzy set (possibility theory) ideas [6,7]
to construct non-linear estimators for
diminishing the
bias comparing to the aforementioned approach. With
biased observations the focus is naturally shifted from
the variance of an estimator, which can be arbitrarily
reduced by increasing a sample size, to its asymptotical
bias. More exactly, in the traditional representation of the
squared standard error (SE)

22
ˆ
ˆˆ
VarEB


our primary point of interest is the first term. Thus, we
start with analyzing the asymptotical bias of the sug-
gested estimators and then compare it to that of estima-
tors traditionally used in statistics such as weighted mean
or weighted median. Then, SE is addressed for small
samples via Monte Carlo simulations when the second
term is not negligible.
Worthy noting that in the simplest situation with unbi-
ased observations 0
j
b
and known
’s the unbiased
estimator with the smallest SE (least square estimate) is
given by the weighted mean, e.g. [8]
22
211 2
22
12
ˆLS
x
x


(2)
where
j
x
is the sample mean of the j-th sample. More-
over, with normal noises it is the maximum likelihood
estimator [1].
C
opyright © 2013 SciRes. AM
L. I. PITERBARG
1270
In the general formulation (1) with biased observations
a choice of an appropriate measure of the estimation skill
is a challenge because the bias ˆ
B
depends on unknown
(nuisance) parameters b1, b2 which never can be identi-
fied from the available observations. We construct such a
measure as follows.
For large samples one can efficiently estimate 2
j
,
and the bias difference 12
from the
observations (1) by subtracting the second sample from
the first one. Introduce
1, 2jbbb 
1
b
b
(3)
Assume that we deal with a class of estimators ˆ
for
which the asymptotical bias exists and
thereby is a function of all the involved parameters
12
,nn

22
12
;,,Bb

(sub ˆ
is dropped as a matter of
breivity).
Unlike 22
12
,,b
one cannot estimate
all un-
der the given observations. In such a situation one of the
ways to order estimators according to their biases is to
accept that

1
ˆ
at
is better than

ˆ2
(under the identifi-
able parameters being fixed) if




12
πBBπ (4)
for a certain class of densities

π
where




22
12
π;,, πd
kk
BBb
 


and is the asymptotical bias of

k
B

ˆk
.
In simple words under arbitrary distribution of
the
absolute value of the first bias is not greater than that of
the second one.
Our first finding and pretty surprising one is that the
best estimator in sense (4) among a wide class including
traditional statistical and fuzzy estimators is the simple
arithmetic mean
12
0
ˆ
2
(5)
where
j
is any consistent estimator of the center of xi
(i.e.
j
j
b
 as ) say the sample mean or
median.
j
n
Does it mean that (5) is the best way to deal with bi-
ased observations? Of course it is not because after all a
real matter of concern is SE which can be essentially less
for the weighted mean (2) than for (5) under small
enough biases
j
b.
The next important result of this study is that the sug-
gested fuzzy estimators are better than weighted estima-
tors of type (2) in sense (4). Finally, to decide which es-
timator should be prefered in dealing with small samples
we carry out Monte Carlo simulations and use SE aver-
aged over the nuisance parameters as a measure of the
estimation skill
 
2
,
ˆˆ
SE E

 (6)
where the angle brackets mean averaging over parame-
ters
defined in (3) and the ratio
1
12
(7)
characterizing the difference in the noise level of two
sources. The reason for including the averaging over
identifiable parameter
is that we are interested in
small samples
1
12 for which it is not possible
to efficiently estimate even identifiable parameters.
, 0nn
We then investigate dependence of (6) on the bias
level b and noise scale 12

 for different esti-
mators and suggest recommendations for sensible choice
among them. In general, fuzzy estimators seem to be
preferable for high values of b and
in most of sce-
narios determined by different noise distribution (normal,
Cauchy), non-probabilistic noise (logistic chaos), and by
different estimates of the center (mean or median).
2. Estimators and Their Asymptotical Bias
First recall that a fuzzy set is a pair
,
A
P where A is a
set and
:0,PA1 is called the membership function,
the value
Px characterizes the degree of membership
x in A. The set
>0PxxA is called the support of
,
A
P. For our purposes it is enough to consider real
fuzzy sets,
A
R.
Regarding to the formulation (1) we consider A as a
range of the observed random variable X. Let us intro-
duce a class of fuzzy estimators similar to estimators
based on the triangular membership function (possibility
distribution) discussed in [9].
Let
F
x be a cumulative distribution function,
symmetric and increasing, i.e. ,
 
1FxF x
Fx
0.
Introduce the membership functions generated by each
of two samples as follows

2i
21if
jf
j
j
j
j
j
j
x
Fx
s
Px
x
Fx
s












and
j
,
j
s
are consistent estimators of the center and
spread of the j-th sample respectively, . In other
words we assume
1, 2j
,
j
jj
bs j


with probability 1 as .
j
n
Copyright © 2013 SciRes. AM
L. I. PITERBARG
Copyright © 2013 SciRes. AM
1271
mean
0p
and the weighted estimator with 1p
. We address the following estimator of
henceforth
called the fuzzy estimator To analyze properties of ˆ
f
, let us fix the bias differ-
ence 12
bb b
, the observation error level 12

and introduce dimensionless parameters 1
bb
,
1

, ab
 . Then the bias of (9) as
12
nn n
 goes to
 

 
12
12
d
ˆ
d
O
f
O
x
Px Px x
PxP xx
(8)
where O is the set of the Pareto optimal solutions and
is the minimum of a and b. Pareto set is defined
by
ab
x
O if and only if


 
,
1
p
ppp
p
p
Bb
  
  (10)
The asymptotical bias for (8) is given in the next
statement.
''
12
<0PxPx
Proposition 1
In other words, a Pareto optimal solution (Pareto op-
timal) is one in which any improvement of one objective
function in the two objective optimization problem
Let 12
nn n

ˆf
, then for any fixed a the bias
ˆf
BE
goes to

f
Bb
  (11)
 
12
max, maxPx Px
can be achieved only at the expense of another, e.g. [10]. where
A key point in the base of (8) is that we use a standard
fuzzy logic aggregation procedure, but integration is car-
ried out only over the set of x maximizing both member-
ships of x. Such an approach goes far beyond linear esti-
mators traditionally used in statistics. However, the in-
troduced class of estimators allows their analytical study
at least regarding to the asymptotical bias.

 
 

11
1
ap q q
ap p

 
 

with
 
 
2
d,
d
a
ax
a
ax
pxxFu u
qxxFu u
Together with (8) we consider the class of weighted
estimators Proof
1
ˆ1
pp p
ww With no loss of generality assume 12
<
, then the
Pareto optimal set is simply an interval
12
,O
due to the condition
>0Fx
1
Px
. Since the symmetry the
only intersection point of and in this
interval is given by

2
Px
2
 
 (9)
where
j
is a sample mean or other consistent estima-
tor of the center of j-th sample.
The weights in (9) are given by

21 2
p
pp
p
wsss 12 21
0
12
s
s
xss
and our primary focus is on the minimum least square
estimator corresponding to , simple arithmetic
2pand after an appropriate variable change (8) becomes













12 12211
12221 12
12 12211
12221 12
22 2111
21
d1
ˆ
d1d
ss s
sss
fss s
sss
d
s
suFu ussuFuu
sFuus Fuu
 
 
 
 


 

 

 
 


Let us change u to u in the second integrals on the top and at the bottom. Then using the symmetry of
F
u get











12 12121
12212 12
12 12211
1222112
22 2111
21
dd
ˆ
dd
ss s
sss
fss s
sss
s
suFu ussuFuu
sFuusFuu
 
 
 
 


 

 

 



Proceeding to the limit , accounting for the consistency of estimates
12
,nn ,
j
j
s
and symmetry of
F
x
obtain








12 12
21
12 12
21
22211 1
21
dd
dd
bb
bb
fbb
bb
buFuu buFu
B
Fu uFu u
 

 

 

 

 

 


u
L. I. PITERBARG
1272
Notice that in the considered case . Using di-
mensionless variables
0b
1
bb
, 1

,
ab
 after some algebra arrive at (11).
Notice that

1xx

1 (12)
Proposition 2
If 0< <12x, then

12x
(13)
Proof
Since inequality (13) is equivalent to

0px
 

2121apx qxapx qx
The expression on the right hand side can be written as
 

12
2d
1d
22
a
ax
ax
a
SxxaxuFu u
va
vF v
x




then the inequality takes form and is
equivalent to
 
>1Sx Sx
 

122 1
d
221
ax ax
aa
va va
vFv vFv
x





 
 

d
x
x
(14)
Let us break down the integral on LHS into integrals
from a to 0 and from 0 to respectively,
change v to v in the second integral, and move it to RHS.
Next, make the same variable change on the RHS of (14)
and move it to LHS. The goal is to have all the integrals
over positive intervals. The result is
12a



12
012
0
dd
22
>d
2
ax a
ax
a
va va
vFv vFv
xx
va
vF v
x





 
 




1
Obviously
 
22va xva x and

21 2vaxva x  whenever 01x2
and hence the last inequality is true since
F
u is in-
creasing. The proof is over.
Further we restrict ourselves by distributions which
decay fast enough as
u

C
Fu u
(15)
From (15) it follows that
 
0
lim 0
xqx px
and
since we get
 
11pq0
00,1

1 (16)
Next we consider the class of all estimators for which
the asymptotical bias is expressible in form


Bb

 
x
satisfies
(17)
where
 
11.
01,0 12 if 012
xx
xx



 
Notice that both the weighted mean (9) and e fuzzy
estimator (8) are in this class. For the former it is easy to
ch
th
eck and for the latter the statement follows from (12,
13, 16).
Now we intend to order estimators from this class ac-
cording to their asymptotical bias. Roughly speaking for
fixed a and
one estimator is better than another if
the asymptotical bias of the former averaged over
is
less than that of the latter. Rigorously, for two estimator
with biases

kk
BB
given by (17)

1, 2kthe
first one is better than the second one if (3) holds true for
any positive fu
nction
π
x
satisfying two conditions
 
πd, π0.5 π0.5
x
xxx x


(18)
Value 0.5
of bala
ition 3
is singled out since it corresp
the case nced biases
onds to
12
0bb.
Propos
If for some 012

inequality
12

he second one is estimato true, then the first r is better than t


12
ππB


Proof
By direct computations with usin
ob
B (19)
g symmetry of π
tain for 0.5z
 
1
dπdπd<0
d
z
xxxx

z
xz
z 

Hence
12
 
implies (19) because
j

0.5.
ising result is rA surpreadily derived from Proposition
3.
Corolla ry
The trivial estimator
12
0
ˆ
2
(20)
is the best estimator in the class (17
This statement follows from the fact that for
)
(20)
012xx

.
The problem with (20) is that for small biases it is es-
han ˆ
sentially less efficient t2
and ˆ
1
. Let us compare
the efficiency of (20) to that of 2
ˆ
in the case of equal
small biases 12
bb
. Introduce ratio of MSEs


2
0
ˆ
E
2
2
ˆ
R
E
Proposition 4
Copyright © 2013 SciRes. AM
L. I. PITERBARG 1273
For any large
(21)
whenever
>0C
RC
11
 111 1
or
2 122 1
CC
CC


(22)
and
2






2
22
222
2
2
22
141
41 1
C
C


 

(23)
oof
Let
Pr


2
214s

 and
 
2
2
22
11t


2
then (21) becomes
2
2
C
2
s
t
or
2
2
21
s
tC
C
(24)
Inequality is equivalent to (23) and then
turns in
xample assume C = 2, then (20) holds true if β >
0.7887 or β < 0.2113. Then for β =
< 0.2594.
orth to co
20stC
(22). (24)
For e
0.1 we have R > 2 if ε
Thus it is wmpare the suggested fuzzy esti-
mators with 2
ˆ
and 1
ˆ
. For that purpose we need an-
other representation for
appearing in (11)
 
 

1TxTx
xx

 (25)
1
apxpx
where
 
2dTxxauFuu
a
ax
Since the denominator in (25) is positiv
following statement gives conditions for for the fuzzy
r to dominate
e for all x the
estimato 1
ˆ
.
Proposition 5
 
>, 0,12xxx
(26)
if and only if
 
1Txx > 0, 0,12T x (27)
e possibility distribuFor example if thtion is given by

1
1
Pu u
(28)
which corresponds to
0.51, <0u, then
 
Fu u
(27) is fulfilled. Indeed, for (28)
2
1l
n1
1
bx
xx
xx

 



Then for any fixed
21
ln1 ,
bx

,1txbTx Tx
1
a
ba
one gets
,0 0tx
0,1 2x
and


12 0
21 1
bx
t
bxbxxbx

which implies (27).
t always fue triangle
bership function defined by the cumulative prob-
ability distribution function
Condition (27) is nolfilled for th
mem

1if
0.5
0if
uqu
q
Fu
uq
0


where q is a fixed quantile. Indeed, direct computations
yield [11].
Proposition 6

 
 









 
32
2
32
2
1231 12
1211
312
11
1ifmin 1,11
3
132
1
ifmax 1,11
12 1
if 111
xxx
xx
xx
x
xx x
xx
x
x
x
x

if 11 1x







 




where

qa
.
be deriIt canved from that statement that (26) holds
true if and only if 12
 while for 173
 the
opposite relation holds true:

x
x
for all 01x2
.
In the intermediate case 73 1
2 the difference
x
x
changes the sign once in

0,1 2. If 1
then two samples are incompatible.
These conclusions are illustrated in Figure 1 by plot-
ting the difference

d
x
xx
for the triangular
membership function (left and middle panel) anr
membership (28) (right panel2
d fo
). If 1
 then for sure
73
and hence
d0x
for 1xt panel). If
23
2 (lef
er
then two options are possible eith
dx
or
0
d
x
changes the sign once (cen
tral pane l). Finally
Copyright © 2013 SciRes. AM
L. I. PITERBARG
Copyright © 2013 SciRes. AM
1274
Figure 1. The difference λ(x) x vs x. 1) Triangle membership, 1 < γ < 2; 2) Triangle membership, 2 < γ < 3; 3) Membership
given by (28), 0.5 a 5.
for membership (2) for all
a.
However, it can be s[9], that

d0x
hown

p
x
x

(29)

2
x
x
when for all

0,1 2x thereby the fuzzy triangle estimator is better
than 2
ˆ
in sense (19).
012x
. Then

1
12
1,12
p
Gxa
 
dx
(29) the region always occupies more
than a half
Nex efine another quantitative for
asy to evaluate,
un he parame
interest
t, dcharacteristic
comparing asymptotical biases which is eThus under G
like

πB. First define the space of tters of . In particular, for

1
x
x
.125 and fo

,1232, 01
 
 
The range for
is founded in [9]. Then introduce
the diffe of biases for the mentioned estimators de-
fined by
thetegral
on the right side is equalr
in
hand to 0
erenc
functions 1

x
and

2
x
respectively

12
,BBB


and define
 

12 , 0GB
 
  ,
Obviously the area of equals 2 a hence the first
estimator can be viewebetter than the second one
he area of exceeds 1. This excess is
quantified in the nextnt.
Proposition 7
d as
12
G
stateme
nd
whenever t
If
 
12
x
x

, 012 then x
 

12
1212
0
11 d1Gxxx

 
Indeed, the solution of

12
<
x
x
 
is
equivalent to
 

12
>2xx
 
from which the
stent readily follows.
The following proposition gives a rough estimate of
improving a fuzzy estimator comparing to any weighted
m
atem
ean (9). Set
 

, ,GB,
fp
B


Corolla ry
Let for some fixed a
 
2
2
2
2
1
x
xxx
 it is ln240.1733. The
latter follows from
 
12
2
0d1lxx

rticular, the area of a region in where the
bias of the fuzzy estimator (28) is less thaat of
n24
In pa
n 1
ˆ
th is
greater than 1.125. While the area of a on in regi
wathere the fuzzy triangle estimor is better than 2
ˆ
is
greater than 1733. 1.
3. Simulations
The goal of the performed simulations is to compare the
efficiency of different estimators underfferent noise
distributions.
di
Now the asymptotical bias is not of primary
concern, but rather the relative standard error
 
2
1.5 1
20.5 0
1
ˆˆdd
2
SE E
 


is addressed under small samples.
o fuzzy estimators (8), first, deter-
1) and, second, determined by
We examine tw
mined by (28) (Fuzzy

0.5eu
Pu
(Fuzzy 2) and three weighted estimators
0
ˆ
, 1
ˆ
and 2
ˆ
given by (9).
Sample sizes 12
10nn
and number of Monte
Carlo trials M = 100 for each mesh

,
are kept the
same for all ex
First, normal noise is tested with
periments.
j
j
x
, i.e. the
L. I. PITERBARG 1275
sample center is estimated by the sample mean (Figure
2).
At zero bias (left panel) 2
ˆ
is best as expected, while
modest biases (central
pa
Fuzzy 2 is only slightly worse. For
nel) 0
ˆ
is best for small level of noise while for
modest and high level of noise again Fuzzy 2 is better.
h biases (right panel) only two estimates
peting ˆ
Finally hig
are co,
for
m0
and Fuzzy 1. The former is better for
small noise while the latter dominates for high noise. 2
ˆ
and 1
ˆ
are much worse in this case.
If the center is estimated by
the sample median the
coe, how
n).
stri
nclusions remain basically the samever the
numbers are slightly different (experiments are not
show
In the case of Cauchy dibution of the noises (Fig-
ure 3) only the median

med
j
j
x
was used for
estimati the centers.
One can see that even for zero bias of noises Fuzzy 2
has the smallest SE along with 1
ˆ
ng
. For modest biases all
the estimators except 0
ˆ
are approximately of the same
accuracy for the whole range of
. The primitive 0
ˆ
appe to be essentially worse than aarsny other estim ate
for high values of
. Finally for high biases again
Fuzzy 1 is best for intermediate values of
while 0
ˆ
and Fuzzy 2 are slightly better for small and large
’s
respectively. Comparing to the normal noise the accuracy
of all the estimators are somewhat lower.
In Figure 4 results are presented for logistic noise
generated by

11
kk
rk


with r = 5.2 for the first sample and r = 3 for the second
one. The results are closer to the normal noise case rather
than to the Cauchy case, however errors in general are
smaller than in the normal case.
In summary for all three experiments, the classical
weighted estimator 2
ˆ
is worsening fast as the level of
bias is increasing while the fuzzy estimators demonstrate
a steady skill in all the scenarios and for whole range of
and b
. At this point it is hard to give a preference
to either Fuzzy 1 or Fuzzy 2.
Similar conclusions can be drawn from Figures 5 and
6 where other experiments are presented in which the
dependence of SE on b were studied for fixed
. We
Figure 2. Dependence of standard error on the noise level σ for different values of bias scale b normal noise. The mean is
taken as an estimate of center. 1) b = 0; 2) b = 0.5; 3) b = 1.
Figure 3. Dependence of standard error on the noise level σ for different values of bias scale b Cauchy noise. The median is
taken as an estimate of center. 1) b = 0; 2) b = 0.5; 3) b = 1.
Copyright © 2013 SciRes. AM
L. I. PITERBARG
1276
Figure 4. Dependence of standard error on the noise level σ for different values of bias scale b logistic noise. The median is
taken as an estimate of center. 1) b = 0; 2) b = 0.5; 3) b = 1.
Figure 5. Dependence of standard error on the bias scale b for different values of noise level σ normal noise. The median is
taken as an estimate of center. 1) σ = 0.1; 2) σ = 1; 3) σ = 3.
Figure 6. Dependence of standard error on the bias scale b for different values of noise level σ Cauchy noise. The median is
taken as an estimate of center. 1) σ = 0.1; 2) σ = 1; 3) σ = 3.
Copyright © 2013 SciRes. AM
L. I. PITERBARG
Copyright © 2013 SciRes. AM
1277
want to stress two of them, first, for high
e and, second,
Fuzzy 2 is
uniformly better than any other estimat0
ˆ
is not appropriate for such values of
regardless
noise distribution.
4. Conclusions and Discussion
A majority of studies in estimating a location parameter
address, first, linear functionals of either the original
sample or its ranking, e.g. [11], and, second, unbiased
observations. Here a class of essentially nonlinear esti-
mators is suggested based on the fuzzy set theory ideas to
handle biased observations coming from two different
sources. Because any analytical investigation of the
standard error for highly non-linear functions of sample
an the classical least square estimator.
r
are estimator and
N00014-11-1-0369 and NSF under grant CMG-1025453
is greatly appreciated.
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5. Acknowledgements
The support of the Office of Naval Research under grant
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