The George Washington. University,. Washington,. D.C.. 20052. ... Osborne. (1942), Cochran. (1946), Mate/m. (1947), and Yates (1949). Recent references are.
OF RESEARCH
BUREAU
STATISTICAL SRD
Research AN
Report
INVESTIGATION VARIANCE FOR by
Chief,
This paper Statistical
SERIES
Number:CENSUS/SRD/RR-84-02 OF SOME ESTIMATORS SYSTEMATIC SAMPLING
Kirk
Statistical U.S. Bureau
was submitted Association
THE CENSUS DIVISION REPORT
M.
OF
Wolter
Research Division of the Census
for publication in in December 1983. ,.
Journal
of
the
American
.
December
AN
12,
1983
INVESTIGATION
OF SOME ESTIMATORS OF VARIANCE SAMPLING
Kirk
M.
FOR SYSTEMATIC
Walter*
ABSTRACT
This paper provides the survey practitioner some guidance about the problem of variance estimation fur equal probability singleAn i o
if
;k
< 0
Based on splitting the sample into p systematic subsamples. When f is negligible, Bias {v,} = (Varf&J -pVar tY1 )/b-l 1 implying that v7 is unbiased when the variance is inversely proportional to sample size. See Koop (1971) regarding the case p = 2.
Devised from a superpopulation model where the correlation between two units in the population depends only on the distance between them. See Cochran (1946), Osborne e.g., (1942), Matgrn (1947). In v8 is an estimator of the 'k correlation between units k units apart. Heilbron (1978) gives three estimtors that are similar to v8.
Table
1.
Eight
Estimators
of
Variance
for
Systematic
Sampling
continued
NOTE: n ' (Yij -j)2/(n-l) j=l n E ij-S)(Yi,j-l-.?)/(n-l)s2 J*= 2 (y
S2
'k
aij b
ij
yij
- yi,j-l
yij
- 2Y i,j-1
+ yi,j-2
'ij
Yij/'
- Yi,j-1
+ Yi,j-2
d
Yi j/2
-
+
ij
9, f
Yi,j-1
-
l
**
- Yi,j-3
’
sample mean of the a-th n/p (where p and n/p are n/N = l/k.
+
Yi,j-Ji2
Yi,j-8/2
systematic integers)
subsample
._
.-
I.
-..
of
size
--
..--
--
I
.
Some
3. In this and
use
it
variance. to
the
ij
as
a criterion
We
assume
=
the
conditions
on
Simple
introduce
for
the
comparing
finite
the
Models
notion
of
various
population
is
model
bias
estimators
generated
of
according
(3.1)
eij,
fixed
constants
variables.
Our
and
under
Pij
on
model
denote
random
Based
shall
the
+
~ij
~ij
a')
(0s
we
section,
superpopulation Y
where
Comparisons
eij
and
main
the
goal
which
is
the
errors the
eij
are
determination
eight
estimators
of of
l
variance
perform
The vu
well
expected
with
bias
((x=1, . . . . 8)
are
respect
and
expected
defined
Brval
=
gE{va}
k?{val
=
&va}/CvartY},
to
model
bias.
relative
bias
of
an
estimator
by
- &ar{g}
and
respectively.
In
&‘a)
for
five
extend
simply
useful
to
In addition 3.5
the
of
for
the
model,
200
finite
and
each
models
of and
as
contained
the
well true
the they
the the
present
form
work, Carlo
of
study
bias
and
mean.
N = MSE
of
variances.
that
made
1000
was models
Table were
each
confidence
These
results
in
in
of
for
describe
Seven
described size
These
error
we
estimators. are
expressions
(3.1).
heterogeneous
proportion
population
we
the
Monte
populations
as
of
analytical
a small
study
- 3.4
with
this
population,
computed,
3.1
model
properties
chosen
in
the to
results
investigate
Sections
Section
were
5.
For
each
generated,
estimator intervals
quantities
to
were
were that
_
9
averaged
over
the
200
populations,
expected
MSE,
and
the
expected
eight
The
estimators.
intervals
was
Linear
B.
where
B. and
errors
eij
variance
are
where
2 of
we
result This
From intercept variance slight
-
in
for
forming
standard
bias,
each
the
normal
the
of
the
confidence distri-
p = 2.
fixed
'
is
the
2.
The
of
the
was
also
assumed
unknown)
by
of
the
the
expected
(3.3)
for
are
was
function
2
v8b
and
given
derived
In
involving
the
by ,
deriving
probability
in by
;,s2)
¶
&!E{iks'}
v8.
> 0 with terms
the
(iid)
variance
&{s2]
that
and
.
of
notation
that
that
for &E{v~}
expectations
guarantees
seen
(1-f)a2/n.
expression
expanded
constants distributed
easily
estimators
expectation
an
this
one.
operator
defined. 2 and
B. has estimators,
effect.
t
eight
used
Table
represented
identically
B2(k2 -1)/12 l of
well
(but
'
model
the
be
(3.2)
is
Table
may
B,[i+(j-l)k],
have
are
trend
It
=
assumption
Xn(*)
level
with
variables.
function
it
studied
and
expectations
same
the
independent
this
approximating the
+
random for
column
of
linear
13, denote
.23artPl The
was
with
=
Pij
u2)
point
used
expected
Trend
Populations
(0s
v7
the
confidence
multiplier
0.025
Estimator
bution.
3.1
the
giving
no
(3.3), effect while
Similarly,
we on
conclude the
that
relative
the
error
variance
the
value
of
the
the
biases a2 slope
value of has Bl
of
the
the only has
a
little
10
effect
on
small.
the
For
close
to
0,
relative
bias,
unless
B,
k is
large
populations
where
we
following
have
the
is
extraordinarily
and
useful
B,
is
not
extremely
approximations:
n -(n-6)/n -(n-6)/n -1 -1 -1 P* from
Thus, and
v3
the
are
The
reader (1977),
trend.
The
linear
trend,
linear
trend
variance
from
is
now
each
of
relative
recall
that
these
suggests
v4
will who
contrasts
view
defining
whereas is
not
v2,
view
of
for ~4,
v3,
and
a desirable
a function
Stratification We
of
bias,
the
estimators
v2
preferred.
Cochran
3.2
point
of
the
results
populations v5,
v8
and
do
v6
not.
property
here
differ
from
with
linear
eliminate
the
Eliminating because
the
the
trend.
Effects the
systematic
n strata.
This
sample situation
as may
a selection be
of
represented
one by
unit the
model ~ij
for That
all is,
=
i and
j,
the
unit
(3.4)
Pj,
where means
the
errors
pij
are
e ij
are
constant
iid within
(0,~')
random
a stratum
variables. of
k units.
0 c b c
. cu
N >
0 +
c . c.3
r-l z
2
Y d
b .
+” h . Y m ,” 12
+ h
.
*
. z
N N .
F
:
c . N D + d -
LA Y co
N b, + hco
: --
+”
.
N
co +
N + m
:
v) . ;r :
0” (u
24 m
Y N
N
.
-7 c. . I
+ + -7 II I N . ‘7 s v
tn 2.
u
4
A
7
,”
-w
w
0 v Y
h Y Y
12
1
.
‘, 13
For
this
class
of
populations,
we
note
=
gi
‘j . - Y . .
and
it
follows
that
the
expected
The
expectaions
column
3 of
expectation is
large
of
Table
2.
of
is
vB Pr{Pk
and
From
Table
estimators
when
stratum
stratum
mean
between
the
absolute which n
=
Based v6
trend
in
v3,
=
(3.5)
not
we
of
the
equal,
biases. expected
(3.5)
variance
expression and
see
that
and
roughly
are
and
is
are
for
will
given
in
the
be
valid
when
n
the
first
seven
1.
small uJ
j
,
of
equal
approximately
there VI
each
can
and
This
vB
striking
often
have
is
for
equal.
be
point
biases
relative
Pj
When
the
differences the
largest
demonstrated . = JI
bias
an(j)
in + sin
.
Table
3
(j)
with
v5,
and
p = 2. on
these
the
contrasts
expected v2,
are
the
provide
The
-> 0)
means
estimators again,
of
. .
(l-f)u2/n.
approximation,
has
relative
and
an
estimators
gives
= 20
eight Once
2 and
variance the
the
- 6
l
variance
VarI?}
*
that
the
most
in
stratum
v7
do
examples,
protection
used
variance and
simple
is not
these means, not
we
against
is
a function
eliminate
the
that
stratification
estimators which
conclude
tend
to
desirable of
such
trend.
v4,
effects.
eliminate here
a trend. Estimators
a linear
because
the
Conversely, v5
and
v6
14
Table Estimators Effects
3. Expected of Variance
Relative Bias Times for Populations with
u2 for Eight Stratification
Estimator %n(j)+sin(j)
j
NOTE:
v1
35.00
0.965
v2
0.50
0.235
v3
0.50
0.243
v4
0.00
0.073
v5
0.00
0.034
v6
0.00
0.013
v7 .
5.00
0.'206
'8
-0.67
n = 20,
p =
2,
u2
= 100.
-0.373
i
15
will
be
preferred When
means. pairs
of
v3
v7
in
and
the
We
are
note
mators
3.3
of
the
the
are
in
is
summable,
has
class We
the
. . . . kn,
The
and
expected
the
smallest
expected
of is Uij bias
ct
where
bias of
when
bias.
the
~j For
p strata.
are p = 2,
bias. a special = IJ for of
the
populations study
series
=
the
are
such
case
of
the
all
i and
j.
first
seven
esti-
For
this
C J-=-OD
Qj
populations
where by
model
(I-f)(l/n){v(O)
't-j
the
unit
assuming
(3.6)
{ajl
sequence
+ ,&
occurs
specification
uncorrelated
for
=
of
may
time
variance
EVar{y}
where
have
Populathons
correlated.
t = 1,
stratum
zero.
Yt'P
for
the
nonoverlapping
groups
expected
in
adjacent
expected
with
trend
in
model
model
important
y-variable
will
terms
random
the
variance
Another values
comparable
case,
Correlated
v3
equal
smallest
effects
special
a nonlinear
nonoverlapping
that
stratifiction this
have
will
is
FLj are
estimator
adjacent
v7
there
means
strata,
Estimator equal
when
(0,~~)
is
absolutely
random
variables.
is
- k&&-
n-l hf,
(n-wf(wl~
kn-1 h$ =
(kn-h)y(h)
(3.7)
16
OD r(h)
By
assuming
moving and
that
average
study For
~(yt-~)(yt-h-v)}
=
(3.6)
arises
process,
their
=
we
from
may
-2
ajaj-hu2*
a low
order
construct
autoregressive,
estimators
of
Var{gsy}
properties.
example,
representation
we for
consider this
the
model
model
is
that
the
first
motivates order
v8.
A
autoregressive
process
Yp
where
p is
the
distinguished O. ..a.:. .* 0.834tOO,1',' !&.796tOO. i,l, 0.755tOO..I:::' '0.74Ot(jO"j:'0.84l+oO " -0,194too : '0.'957+00 ' 0.834tFO ....',.? ; .%0.281 tO0 : ,'-0.853-01 0.282tOO 0.281tOO 0.281tOO':~;~ (1.28OtOO ";0,28QtOO ~:~~~;~"0.28OtOO !
0.741-01
0,741.01~
0.741.oy..%i 0,741-01,':~~~ 0.741-01 ~:::'~O.-rsl-oi :;;;: 0,741-01 ': -0.292-01
0.207+01 .' ';'.. OJfjfjtO1:',i',".'O.j5O&J ',': :::. 0.590#1 0.104,t!.Ie0.293tOl ': 0,293+01 :':;;j
'0.427tOl : 0.243#1 ' 0.112tOl 0.118t.03 0.533+02
:, -0:200tOO
0,20$tOl ::'. 0.174$1:;~;.: 0',]63tO1 .';i.; 0.29kt81 0.243tOl,.,; ""..;
-0.907-01
0.103+01 "
0.103#i~~:~~.~~,0.100#1 ....:.. 0.974t-00 ;;:'JL961400 : O,104to'l*. -0.321-01
0.370t01
0.37OtOi')“;~;' 0.220i01 '.:..i 0.174tO;]$ 0.15'6+01' .'~'~;'O .599402
,-0.200+00
0.419101. ~0.419tOl';:c'i~! 0.263tOl.::' 0.211#1 '~::~,0.196~1 % 0.275+02.''.-0.909-01 .8.. ,11 0.183#2 . 0.402Ml 0,402tOl'.:T,;,j 0,278+01'1.1,. 0.225tOj,.'..'.. O.lOltO2 -0.323-01 .;'0.205-H)1 ,,:
NOTE: Resultsignoreterms of order no2.
20
3.5
Monte The
Monte
presented basis in
Carlo
Carlo
in
of
the
this
terms
Results
of
row
both
that
seriously
on
this
is
For the
v1
to
be
although
more
than
of
is
for
bias,
v7
with
and
linear The
A4
is
permit
rate
of
because
the
95
this
autocorrelated
actual
On
variance of
and
freedom",
was
actual in
all
cases.
A2),
all
of
particularly
levels
good
vB
are
performance.
constructed Because
unattractive
and
estimators,
The
populations.
choice
estimator
confidence
percent.
the
95
only
labeled
v3,
estimator
particularly
best
rate
remaining
and
the
The
row
v2,
8.
produce
"degrees
(see
and
the
the
nominal
biased.
biased
is
are
for
of
large
populations
.
trend.
Monte
populations
the
populati'on
downward
of
of
to
estimator.
than
than
seems
vB
variance
acceptable
are
vl
7,
ability
number
seriously
surprising
VT
lower
trend
6,
Estimator
the
population
Tables
the
preferred
nominal
specific,ally
of
and
The
to
are
is
the
MSE
the
are
each
lower vB
is
linear
estimators
seem
Al
random
estimator
biased.
levels the
the
intervals.
related
basis
confidence
labeled
minimum
confidence
estimators
for
investigation,
percent is
results
Carlo are
essentially negative
results
presented the
they
preferred
here;
remaining
estimators.
in
have v5
the
stratification
labled
A3
A3,
except
truncated
Estimators
v2,
smaller
absolute
as
and
v6
have
and
effects
rows
same
values.
for
v3,
and bias
equally
A4.
Population so
v4 and
small
as
are
not
to
clearly
MSE bias
than but
the
.-
. . .*
._
:.
.
.: . . ...’
::
: . :
._:
:
‘. ’ : :.:r , ;.:.-+..-
._.
..’ ;
*_.- ,. . . : ,,‘: :- : 1..
I .* :.
‘-
,_
:’ -..: ‘.’
.
__ . .
..
. ‘-
‘. .
..
.,._
..
.:,
..:.,..:
:..,.
.’
:,: ._,
:. =.
‘. _, :. :, : : ._ . _’ ,.‘. . .;’ . .. ..: -.. _ ‘._ ,’ . . :: - ., . . . : 1 :.’ ,I . . .. .‘-
:
..
::..
-.
;...
_ .
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..
-.
:
.
. .
,. ~ :.
._:
..
.
.
:
. . :
.
.
:
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”
”
.
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: :_: _. .‘.
:
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.
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.
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.
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: . ...
.
:
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i-,,_.~
-__ .::e.‘ij’ lid‘ N(OJO0) ,_.;-.-‘50 ,.-i /+ (j-r)k -m.+.,J-lnear Trend 1 :...:.20 1‘ ;: .‘,‘. -.,‘r- *. ._. ..: .I :...?” .,_ -I.“,:,:--;. ‘-,‘.:;-:..‘..... : J... ,,.. g. .+t W,“. .1. ‘. .I *A,..-...-.1-; : ;-:;.. /.
, .“.’.;. ”
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:
.-.;::.::;:?
l-. ‘.-.7. .3 ‘i. :.-.‘...r... ‘t, ;. ..
e...:.
.; .. Rtitocorre~ated ‘“. 20’. ” 51
‘.
: A6 :
Autocorrel ated
same as AS with p = 0.4
,, eij iidN(O,JOO) ,.. .:. 20; _’.i:50 20 slnC(2;1/50) ....,.; ’ ::-.,._ .-. ( . . .,,i: _‘. i 1. ,_’ . ..[i+&t )k]l .,. :. . o... ; ;._ : :. . .: . -*.z -., ;.. .: 1..: . : ._ .-. .._: ‘. ‘. ..i ... -.-,-;._ ‘. . . .: .*.,-; i .....rz ..-d .! ,.,.,. 1, ‘.> ..f’ _ .. . .‘_ .:i.*=; ;;-; ,: -7 .*.:..;, ) .. “.!,,-i, ‘. .;. ; _ .: --./.-. :: .~’..f _a : .. i: .,.., -_.-.. : ,: ._. ..
;
22
larger
variance,
“degrees
of
presumably
freedom."
estimators
vl,
v7,
stratification for
and
the
Estimator
v8
(A5),
population
(A6).
the
confidence
of
the
v8
a deficiency
because
large
of
unattractive
are
first
autocorrelated
p erforms
population
actual
Primarily
of
for
in
the
bias,
populations
with
are
A5
effects.
Results
A6.
because
but
not
Even
four
well as
in
in
well
the
level
populations the
in
highly
the
presence
estimators
could
high
with
be
rows
and
autocorrelated
moderately of
associated
in
autocorrelated
autocorrelation,
v8
is
low.
recommended
for
Any low
one
auto-
correlation. Row
A7
gives
the
periodic
population.
interval
k = 50.is
As
are
confidence
intervals
series
badly
programming generated
are
precision,
IMSL
giving
using
real Income (CPS).
now
compare
populations. Supplement The
Monte
the
EXEC
FORTRAN
subroutines.
Some
were
V.
decimal
All
the
the
sampli-ng
eight
associated
on
UNIVAC
system.
random
place
Numerical
The
numbers
were
1100
made
were in
single
accuracy.
Comparisons
The
first
six
consisted
the
performed
8 operating
estimators
populations
the
Computations
March,
of
of
unusable.
eight
the
all and
the
to
study
(because
period)
here
8-9
Carlo
anticipated
downward,
the
about
the
completely
was
4. We
of
presented
language by
to
biased
simulations computers,
was
equal
estimators 0
All
results
of
populations 1981 of
Current all
variance were
using taken
Population
persons
age
eight from
the
Survey 14+,
in
the
23
U.S.
civilian
Standard
labor
force,
Metropolitan EMPINC,
unemployment
indicator
EMPRSA,
Y
for
the
INCNOO,
the
ordered
by
person race
y-variable
by
the
by
age
age
in
natural
customary
ordering.
These
The of
last
6,500
fuel
y-variable ordered
two
was by
State
identification order
was
unemployed
=
0,
if
employed
populations,
before
black
ascending file
CPS
populations
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