ass="wseb v0"> = 10%, 20% and 40%
outliers are indicated by their associated α0 value.
find the value of by using a predetermined , the
probability of having at least good subsamples in .
In view of the breakdown probability function, this
amounts to selecting by requiring 0
k*
p
*
rk
k

*
=1BP p
,
which is a condition imposed on

BP
at a single
point 0
. An alternative way of selecting is to im-
pose a stronger condition on
k

BP
over some interval
of interest.
Note that everything else being equal, we can get a
more robust SUE by using a larger k. For practical
applications, however, we caution that a very large
will compromise both the computational efficiency of the
subsampling algorithm and the efficiency of the com-
bined sample
k
g
S as an estimator of the good data set.
The latter point is due to the fact that in practice, the
subsamples forming
g
S
*
r
are not independent random
samples from the good data set; in the extreme case
where k goes to infinity, the subsample with the smallest
γ-score will appear infinitely many times, and thus all
subsamples in the union of
g
S are repeats of this same
subsample. This leads to the lowest efficiency for
g
S
with
=ESnNk
Fgs . Thus when selecting the
value, it is necessary to balance the robustness and the
efficiency of the SUE.
To conclude Section 3, we note that although the
selection bias problem associated with the combined
sample
g
S can make the asymptotic theory of the SUE
difficult, it has little impact on the breakdown robustness
of the SUE. This is due to the fact that to study the
breakdown of the SUE, we are only concerned with
whether
g
S contains any outliers. As such, the size of
g
S and the possible dependence structure of points
within are irrelevant. Whereas in Section 3.1 we had to
make the strong assumption that
g
S is a random sample
from the good data in order to borrow the asymptotic
theory from the classical method , here we do not
need this assumption. Indeed, as we have pointed out
after (14) that the breakdown results in this section are
valid under only a weak assumption, that is, the
criterion employed is capable of separating good subsam-
ples from subsamples containing one or more arbitrarily
large outliers. Any reasonable should be able to do
so.
4. Applications of the Subsampling Method
In this section, we apply the subsampling method to three
real examples through which we demonstrate its use-
fulness and discuss issues related to its implementation.
For the last example, we also include a small simulation
study on the finite sample behaviour of SUE.
An important issue concerning the implementation of
the subsampling method which we have not considered
in Section 2 is the selection of classical method
and
Copyright © 2012 SciRes. OJS
M. TSAO, X. LING
OJS
289
 
 
Copt © SciR yrigh 2012es.
goodness-of-fit criterion . For linear regression and
non-linear regression models, the least squares estimation
(LSE) method and the mean squared error (MSE) are
good choices for and , respectively, as the LSE
and MSE are very sensitive to outliers in the data.
Outliers will lead to a poor fit by the LSE, resulting in a
large MSE. Thus a small value of the MSE means a good
fit. For logistic regression and Poisson regression models,
the maximum likelihood estimation (MLE) method and
the deviance (DV) can be used as and , respec-
tively. The MLE and DV are also sensitive to outliers. A
good fit should have a small DV. If the ratio
DV e
np
is much larger than 1, then it is not a good fit.
Another important issue is the proper selection of the
working proportion of outliers or equivalently the (esti-
mated) number of outliers in the sample. This is
needed to determine the and to run the subsam-
pling algorithm. Ideally, the selected value should be
slightly above the true number of outliers as this will lead
to a robust and efficient SUE. If we have some infor-
mation about the proportion of outliers in the sample
such as a tight upper bound, we can use this information
to select . In the absence of such information, we may
use several values for to compute the SUE and
identify the most proper value for the data set in question.
For values above the true number of outliers, the
SUE will give consistent estimates for the model para-
meters. Residual plots will also look consistent in terms
of which points on the plots appear to be outliers. We
now further illustrate these issues in the examples below.
Example 2: Linear model for stackloss data
m
k
m
*
r
m
y
The well-known stackloss data from Brownlee [7] has
21 observations on four variables concerning the opera-
tion of a plant for the oxidation of ammonia to nitric acid.
The four variables are stackloss (), air flow rate (1
x
),
cooling water inlet temperature (2
x
) and acid concen-
tration (3
x
). We wish to fit a multiple linear regression
model,
0112233
=yxxx
m
m
 

to this data. We use the LSE and MSE for and
,
respectively, in the SUE. We also try three m values,
and 6, which represent roughly 10%, 20% and
30% working proportion of outliers in the data. The sub-
sample size is chosen to be the default size of s.
The corresponding values for and k in the SAL
and the estimates for regression parameters are given in
Table 2. For comparison, Table 2 also includes the
estimates given by the LSE and MME, a robust estimator
introduced in [6]. The residual versus fitted value plots
for the LSE and SUE are in Figure 5. Since the regres-
sion parameter estimates for the SUE with and
=2
,4m
=11n
*
r
=4m
(a) (b)
(c) (d)
Figure 5. Residual versus fitted value plots for Example 2: (a) LSE; (b) SUE with m = 2; (c) SUE with m = 4; (d) SUE with m
= 6. The dashed lines are ˆ
3, which are used to identify outliers.
M. TSAO, X. LING
290
Table 2. Regression parameter estimates with standard errors in brackets for Example 2.
SUE (m = 2) SUE (m = 4) SUE (m =6)
*=6r*=5r*=4r
parameter LSE MME
=5k=3k=2593k7 27
0
39.92 (11.90) 41.52 (5.30) 38.59 (10.93) 37.65 (4.73) 36.72 (3.65)
0.72 (0.13) 0.94 (0.12) 0.77 (0.13) 0.80 (0.07) 0.84 (0.05)
1
2
1.30 (0.37) 0.58 (0.26) 1.09 (0.35) 0.58 (0.17) 0.45 (0.13)
3
0.15 (0.16) 0.11 (0.07) 0.16 (0.14) 0.07 (0.06) 0.08 (0.05)
3.24 1.91 2.97 1.25 0.93
sample size =2
n=2n=2
e
n=1
e
n=15
e
n
1 1 0 7
=6m
=4
are consistent and the corresponding residual
plots identify the same 4 outliers, m is the most
reasonable choice. The effective sample size for
g
S
=4m=1n
when is e, and hence this 7
g
S
=2
includes
all the good data points in the data and the SUE is the
most efficient. It is clear from Table 2 and Figure 5 that
the LSE and the SUE with fail to identify any
outliers and their estimates are influenced by the outliers.
The robust MME identifies two outliers, and its estimates
for 12
m
,
and 3
are slightly different from those
given by the SUE wi=4. Since the MME is
usually biased in the estimation of the intercept 0
th m
, the
estite of 0
ma
from the MME is quite different. This
data set has been analysed by many statisticians, for
example, Andrews [8], Rousseeuw and Leroy [3] and
Montgomery et al. [9]. Most of these authors concluded
that there are four outliers in the data (observations 1, 3,
4 and 21), which is consistent with the result of the SUE
m
with =4.
m
=6m n
m
=4m
m
Note that the SUE is based on the combined sample
which is a trimmed sample. A large value assumes
more outliers and leads to heavier trimming and hence a
smaller combined sample. This is seen from the SUE
with where the effective sample size e is 15
instead of 17 for . Consequently, the resulting
estimate for the variance is lower than that for .
However, the estimates for the regression parameters
under and are comparable, reflecting the
fact that under certain conditions the SUEs associated
with different parameter settings of SAL algorithm are
all unbiased.
=4
=6=4m
Example 3: Logistic regression for coal miners data
Ashford [10] gives a data set concerning incidents of
severe pneumoconiosis among 371 coal miners. The 371
miners are divided into 8 groups according to the years
of exposure at the coal mine. The values of three vari-
ables, “years” of exposure (denoted by
x
) for each
group, “total number” of miners in each group, and the
number of “severe cases” of pneumoconiosis in each
group, are given in the data set. The response variable of
interest is the proportion of miners who have symptoms
of severe pneumoconiosis (denoted by ). The 8 group
proportions of severe pneumoconiosis are plotted in
Figure 6(a) with each circle representing one group.
Since it is reasonable to assume that the corresponding
number of severe cases for each group is a binomial
random variable, on page 432 of [9] the authors consi-
dered a logistic regression model for , i.e.,
Y
Y



01
01
exp
=.
1exp
x
EY
x



To apply subsampling method for logistic regression,
we choose the MLE method and the deviance DV as
and
, respectively. With groups, we set
and 2 in the computation, and set the subsample size to
. The corresponding values for and k are
=8N=1m
=5
s
n*
r
*,=4,23rk and

*,=3,76rk =1m
=1m
for and 2,
respectively. The original data set has no apparent out-
liers. In order to demonstrate the robustness of the SUE,
we created one outlier group by changing the number of
severe cases for the 27.5 years of exposure group from
original 8 to 18. Consequently, the sample proportion of
severe pneumoconiosis cases for this group has been
changed from the initial 8/48 to 18/48. Outliers such as
this can be caused, for example, by a typo in practice.
The sample proportions with this one outlier are plotted
in Figure 6(b). The regression parameter estimates from
various estimators are given in Table 3, where the M
method is the robust estimator from [11]. The fitted lines
given by the MLE, the SUE and the M method are also
plotted in Figure 6. For both data sets, the SUE with
and 2 gives the same result. The SUE does not
find any outliers for the original data set, while it
correctly identifies the one outlier for the modified data
set. For the original data set, the three methods give
almost the same estimates for the parameters, and this is
reflected by their fitted lines (of proportions) which are
Copyright © 2012 SciRes. OJS
M. TSAO, X. LING 291
(a)
(b)
Figure 6. Sample proportions and fitted proportions for Example 3: (a) The original data set with no outlier group; (b)
Modified data with one outlier group. Note that in (a) the fitted lines are all the same for the MLE, SUE and M methods,
while in (b) they are different.
Table 3. Regression parameter estimates with standard errors in brackets for Example 3.
Original data With one Outlier group
Parameter
MLE MLE M
SUE
4.80 (0.57) 4.07 (0.46) 4.80 (0.59) 5.24 (0.70)
0
1
0.09 (0.02) 0.08 (0.01) 0.09 (0.02) 0.10 (0.02)
nearly the same as can be seen in Figure 6(a). For the
modified data set, the SUE and the M method are robust,
and their fitted lines are in Figure 6(b). The outlier has
little or no influence on these fitted lines.
Example 4: Non-linear model for enzymatic reaction
data
To analyse an enzymatic reaction data set, one of the
models that Bates and Watts [12] considered is the well-
known Michaelis-Menton model, a non-linear model
given by
0
1
=,
x
yx
(16)
where 0
and 1
are regression parameters, and the
error
LSE (dotted) and SUE (solid), and Figure 7(b) shows
the residual plot from the SUE fit. Since there is only one
mild outlier in the data, the estimates from the LSE and
SUE are similar and they are reported in Table 4.
We also use model (16) to conduct a small simulation
study to examine the finite sample distribution of the
SUE. We generate 1000 samples of size where,
for each sample, 10 observations are generated from the
model with 0
=12N
=215
, 1=0.07
and a normally dis-
tributed error with mean 0 and =8
. The other 2
observations are outliers generated from the same model
but with a different error distribution; a normal error
distribution with mean 30
and =1
. The two out-
liers are outliers, and Figure 7(c) shows a typical
sample with different fitted lines for the LSE and SUE.
The estimated parameter values are also reported in
Table 4. For each sample, the SUE estimates are com-
puted with s, and . Figure 8 shows
the histograms for 01
y
=7n*=4r=63k
ˆˆ
ˆ
,,
2
has variance
. The data set has
observations of treated cases. Response variable is
the velocity of an enzymatic reaction and
=12N
y
x
is the
substrate concentration in parts per million. To compute
the SUE, we use the LSE method and the MSE for

and the sample size of
g
S
,,
.
The dotted vertical lines are the true values for 01
and , respectively, and set and s which
lead to and . Figure 7(a) shows the
scatter plot of versus
=2

=10n
0
ˆ
and . Table 5 shows the biases and standard
errors for the LSE and SUE based on the simulation
study. The distributions of the SUE estimators
m=7n
=6
*=4r3k
y
and
x
and the fitted lines for the
Copyright © 2012 SciRes. OJS
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292
(a) (b)
(c) (d)
Figure 7. Fitted regression lines for Example 4: (a) The LSE and SUE lines for the real data set; (b) SUE fit residual plot for
the real data set; (c) The LSE and SUE lines for the simulated data with outliers; (d) SUE fit residual plot for the simulated
data. Dotted lines in plots (b) and (d) are the ˆ
3
lines.
(a) (b)
(c) (d)
ˆ0ˆ1ˆ
Figure 8. Histograms for the SUE: (a) Estimator
; (b) Estimator
; (c) Estimator
; (d) The sample size of
g
S
. The
vertical dotted lines are the true values for 01
,,

n and .
Copyright © 2012 SciRes. OJS
M. TSAO, X. LING 293
Table 4. Regression estimates with the standard errors in brackets for Example 4.
Original Simulated
Data set parameter
LSE (s.e.) SUE (s.e.) LSE (s.e.) SUE (s.e.)
0
212.68 (6.95) 216.62 (4.79) 210.35 (9.03) 217.30 (4.55)
1
0.064 (0.008) 0.072 (0.006) 0.073 (0.014) 0.069 (0.007)
10.93 7.10 15.23 6.47
Table 5. Results for the simulation study.
Parameter LSE bias (s.e.) SUE bias (s.e.)
0
3.64 (8.84) 1.64 (8.93)
1
0.0059 (0.0238) 0.0052 (0.0266)
5.70 (1.51) 0.20 (2.71)
1
ˆ
are approximately normal, and the biases are much
smaller than that of the LSE. That of the estimated
variance also looks like a 2
distribution. The average
effective sample size of
g
S
=10n
*
,,nrk
is 9.93, which is very close
to the number of good data points . There are a
small number of cases where the effective sample size is
12. These are likely cases where the “outliers” generated
are mild or benign outliers and are thus included in the
combined sample.
5. Secondary Criterion and Other Variations
of the Subsampling Algorithm
The 5-step subsampling algorithm SAL (s) intro-
duced in Section 2 is the basic version which is straight-
forward to implement. In this section, we discuss modifi-
cations and variations which can improve its efficiency
and reliability.
5.1. Alternative Stopping Criteria for
Improving the Efficiency of the
Combined Subsample
S
In Step 5 of SAL (), the first subsamples in
*
,,nrk *
r

12 ,
k
s
the sequence
,,
A
AA*
r are identified as
good subsamples and taken union of to form
g
S
*
r
*
,,nrk *
. How-
ever, it is clear from the discussion on parameter selec-
tion in Section 2.2 that there are likely more than
good subsamples among the k generated by SAL
(s). When there are more than r good sub-
samples, we want to use them all to form a larger and
thus more efficient
g
S. We now discuss two alternatives
to the original Step 5 (referred to as Step 5a and Step 5b,
respectively) that can take advantage of the additional
good subsamples.
Step 5a: Suppose there is a cut-off point for the
scores, say C
, such that the jth subsample is good if
and only if

C
j
. Then we define the combined
subsample as



:
=
C
j
g.
j
j
SA

(17)
Step 5b: Instead of a cut-off point, we can use

1
as
a reference point and take the union of all subsamples
whose
scores are comparable to

1
. That is, for a
pre-determined constant >1
, we define the combined
subsample as
 


1
:
=
j
g.
j
j
SA

(18)
In both Steps 5a and 5b, the number of subsamples in
the union are not fixed. The values of C
and
depend on ,
n
s
n,
,
and the underlying model.
Selection of C
and
may be based on the distri-
bution of
-scores of good subsamples.
If either Step 5a or 5b is used instead of the original
Step 5, we need to ensure the number of subsamples
taken union of in (17) or (18) is no less than . If it is
less than , then the criterion based on C
*
r
*
r
or
may
be too restrictive and it is not having the desired effect of
improving the efficiency of the original Step 5. It may
also be that the number of good subsamples is less than
and in this case, a re-run of SAL with a larger is
required.
*
r k
*
r
Finally, noting that the subsamples making up
g
S
*
r
k
,,
in Step 5 may not be distinct, another way to in-
crease efficiency is to use distinct subsamples. The
number of good subsamples in a sequence of distinct
subsamples follows a Hypergeometric (
g
T) dis-
tribution, where
kL L
g
L is the total number of good sub-
samples of size
s
n L
L
and T the total number of sub-
samples of this size. Since T is usually much larger
than
g
L
k
*
r*
p
i
, the hypergeometric distribution is approxi-
mately a binomial distribution. Hence the required
for having good subsamples with probability is
approximately the same as before.
5.2. Consistency of Subsamples and Secondary
Criterion for Improved Reliability of the
Subsampling Algorithm
j
β
and
β
be the estimates given by (method Let
Copyright © 2012 SciRes. OJS
M. TSAO, X. LING
294
applied to) the ith and jth subsamples, respectively. Let

,
ij
dβ
β
be a distance measure. We say that these two
subsamples are inconsistent if c where
c is a fixed positive constant. Conversely, we say that
the two subsamples are consistent if c

,>
ij
ddββ
d

,
ij
dd
ββ .
Inconsistent subsamples may not be pooled into the com-
bined sample
g
S to estimate the unknown
β
.
Step 5 of SAL () relies only on the γ-ordering
of the subsamples to construct the combined sample
*
,,
s
nrk
g
S.
In this and the two variations above,
g
S is the union of
( or a random number of) subsamples with the
smallest γ-scores. However, an
*
r
g
S constructed in this
manner may fail to be a union containing only good
subsample as a small γ-score is not a sufficient condition
for a good subsample. One case of bad subsamples with
small γ-scores we have mentioned previously is that of a
bad subsample consisting of entirely outliers that also
follow the same model as the good data but with a dif-
ferent
β
. In this case, the γ-score of this bad subsample
may be very small. When it is pooled into the
g
S,
outliers will be retained and the resulting SUE will not be
robust. Another case is when a bad subsample consists of
some good data points and some outliers but the model
happens to fit this bad subsample well, resulting in a very
small γ-score. In both cases, the criterion is unable to
identify the bad subsamples but the estimated
β
based
on these bad subsamples can be expected to be incon-
sistent with that given by a good subsample.
To guard against such failures of the criterion, we
use the consistency as a secondary criterion to increase
the reliability of the SAL (). Specifically, we
pool only subsamples that are consistent through a
modified Step 5, Step 5c, given below.
k
*
,,
s
nr
i
Step 5c: Denote by
β
the estimated value of
β
based on ()i
A
where and let =1,2, ,ik
,
ij
dβ
β
be a distance measure between two estimated values.
Take the union

*
=1
=
j
r
gi
j
S
,A (19)
where

j
i
A
(jr
ii ) are the first consis-
tent subsamples satisfying
*
12
=,, ,i i

,
x
jl
ii
jl
iiββ
c
d
*
r
,,,
ma c
dd
for some predetermined constant in

12 k
A
AA
.
The criterion is the primary criterion of the sub-
sampling algorithm as it provides the first ordering of the
subsamples. A secondary criterion divides the γ-
ordered sequence into consistent subsequences and per-
forms a grouping action (instead of an ordering action)
on the subsamples. In principle, we can switch the roles
of the primary and secondary criteria but this may sub-
stantially increase the computational difficulties of the
algorithm. Additional criteria such as the range of the
elements of
β
k
may also be added.
With the secondary criterion, the first subsamples
taken union of in Step 5c has an increased chance of all
being good subsamples. While it is possible that they are
actually all bad subsamples of the same kind (consistent
bad subsamples with small γ-scores), this is improbable
in most applications. Empirical experience suggests that
for suitably chosen metric
*
r

,
ij
dβ
β
, threshold and
a reasonably large subsample size
c
d
s
n=0.5 1nN (say s
),
the first subsamples are usually consistent, making
the consistency check redundant. However, when the
first subsamples are not consistent and in particular
when *
r, it is important to look for an explanation.
Besides a poorly chosen metric or threshold, it is also
possible that the data set actually comes from a mixture
model. Apart from the original Step 5, a secondary criterion
can also be incorporated into Step 5a or Step 5b.
*
r
*
r
*
ir
*=1r
*=1r
k
Finally, there are other variations of the algorithm
which may be computationally more efficient. One such
variation whose efficiency is difficult to measure but is
nevertheless worth mentioning here requires only
good subsample to start with. With , the total
number of subsamples required () is substantially re-
duced, leading to less computation. Once identified, the
one good subsample is used to test points outside of it.
All additional good points identified through the test are
then combined with the good subsample to form a com-
bined sample. The efficiency of such a combined sample
is difficult to measure and it depends on the test. But
computationally, this approach is in general more effi-
cient since the testing step is computationally inexpen-
sive. This approach is equivalent to a partial depth func-
tion based approach proposed in [13].
6. Concluding Remarks
It is of interest to note the connections among our sub-
sampling method, the bootstrap method and the method
of trimming outliers. With the subsampling method, we
substitute analytical treatment of the outliers (such as the
use of the
functions in the M-estimator) with com-
puting power through the elimination of outliers by re-
peated fitting of the model to the subsamples. From this
point of view, our subsampling method and the bootstrap
method share a common spirit of trading analytical
treatment for intensive computing. Nevertheless, our sub-
sampling method is not a bootstrap method as our objec-
tive is to identify and then make use of a single combined
sample instead of making inference based on all boots-
trap samples. The subsampling based elimination of out-
liers is also a generalization of the method of trimming
outliers. Instead of relying on some measure of outlying-
Copyright © 2012 SciRes. OJS
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Copyright © 2012 SciRes. OJS
295
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subsampling method uses a model based trimming by
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instead of some measure of central location and they are
only implicitly identified as being part of the data points
not in the final combined sample.
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The subsampling method has two main advantages: it
is easy to use and it can be used for any regression model
to produce robust estimation for the underlying ideal
model. There are, however, important theoretical issues
that remain to be investigated. These include the charac-
terization of the dependence structure among observa-
tions in the combined sample, the finite sample distri-
bution and the asymptotic distribution of the SUE. Un-
fortunately, there does not seem to be a unified approach
which can be used to tackle these issues for SUEs for all
regression models; the dependence structure and the
distributions of the SUE will depend on the underlying
regression model as well as the method and criterion
chosen. This is yet another similarity to the bootstrap
method in that although the basic ideas of such com-
putation intensive methods have wide applications, there
is no unified theory covering all applications and one
needs to investigate the validity of the methods on a case
by case basis. We have made some progress on the sim-
ple case of a linear model. We continue to examine these
issues for this and other models, and hope to report our
findings once they become available.
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Step GM-Estimates and Stability of Inferences in Linear
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[7] K. A. Brownlee, “Statistical Theory and Methodology in
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[8] D. F. Andrews, “A Robust Method for Multiple Linear
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7. Acknowledgements
[12] D. M. Bates and D. G. Watts, “Nonlinear Regression
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The authors would like to thank Dr. Julie Zhou for her
generous help and support throughout the development
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the National Science and Engineering Research Council
of Canada.
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M. TSAO, X. LING
296
Appendix: Proofs of Theorems 1 and 2
Proof of Theorem 1:
Let
g
p be the probability that random subsample of
size
s
n from
S
>0p
**
,,AA
l
,,AA
>0
*
is a good subsample containing no
outliers. Since , we have .
s g
With probability 1, the subsequence 12 is an
infinite sequence. This is so because the event that this
subsequence contains only finitely many subsamples is
equivalent to the event that there exists a finite such
that 1ll contains no good subsamples. Since
g, the probability of this latter event and hence that
of the former event are both zero. Further, under the
condition that
nn
p
j
n
A
S, *
j
A
may be viewed as a ran-
dom sample of size
s
n
**
,,AA
taken directly without replace-
ment from n. Hence with probability 1, 12 is
an infinite sequence of random samples from the finite
population n. It follows that for any
S
S
j
n
zS
, the
probability that it is in at least one of the *
i
A
is 1. Hence
n. This and the fact that imply
the theorem.
PS B
n
BS
*
=BA
=1
Proof of Theorem 2:
We prove (4) by induction.
For , since 11
which contains exactly =1j
s
n
points, is a constant and
1
W

1
==.
1
s
F
EB EW n
nn
=1j
E
Hence (4) is true for .
To find 2F, denote by
B*
1
A
the complement of
*
1
A
with respect to n. Then S*
1
A
contains
s
nn
points. Since 2
*
A
is a random sample of size
s
n
S
taken
without replacement from n, we may assume it
contains 1 data points from U*
1
A
and 1s
nU
data
points from *
1
A
. It follows that has a hypergeo-
metric distribution
1
U

,,,
ss
nnnn
1HypergU (20)
with expected value

1
EU =.
s
s
nn
nn



**
21
=BAA
1
.U
 
21
2
==
=1 .
s
sss
s
nn
EWnEUn nn
nn
nn












Since 2
, its number of data points
Hence
2
=
s
Wn
It follows that
2
2=1,
s
F
nn
EB n



=2j

12j
j

#
and formula (4) is also true for .
Now assume (4) holds for some . We show
that it also holds for . Denote by
A
the number
of points in
A
. Then
*
=# =#=
11
1
j
jjjjj
WB BAWU
, where
 1
U
j
is
the number of good data points in
j
B1
but not in B
j
.
The distribution of
j
U1
conditioning on 11
=
j
j
Ww
is
 
11 11
=Hyperg,,,
jj jjs
UW wnnwn
 
1j
(21)
By (21) and the assumption that (4) holds for
,
we have






111
1
1
1
1
1
=
=
=
=1
==1.
jj jj
j
js
s
sj
j
s
ss
jj
ss
j
EWEWEEUW
nW
EWE nn
nn
nEW
n
nn
nnn n
nn nn
nn
n
n

 







 














It follows that
j
==1 .
js
Fj
EW nn
EB nn



j
Thus (4) also holds for , which proves Theorem 2.
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