Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
Sparse and Efficient Replication Variance
Estimation for Complex Surveys
Jae Kwang Kim1
Iowa State University
April 4th, 2012
Westat
1
Joint work with Changbao Wu at University of Waterloo
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
1
Introduction
2
Fully Efficient Replication Weights
3
Sparse and Efficient Replication Weights
4
Balanced Sampling
5
Simulation Studies
6
Concluding Remarks
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
1
Introduction
2
Fully Efficient Replication Weights
3
Sparse and Efficient Replication Weights
4
Balanced Sampling
5
Simulation Studies
6
Concluding Remarks
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
Variance Estimation
Construct confidence intervals
Conduct hypothesis tests
Provide descriptive measures on accuracies of estimates
Measure the efficiency of the sampling design
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
Traditional Approaches
Inclusion probabilities πi = P(i ∈ S) and πij = P(i, j ∈ S)
Basic design weights wi = 1/πi
The Horvitz-Thompson estimator of Y =
X
wi yi .
Ŷ =
PN
i=1 yi
(1)
i∈S
Variance estimation
V̂ =
XX
Ωij yi yj ,
(2)
i∈S j∈S
where Ωij = (πij − πi πj )/(πij πi πj ).
Nonlinear statistics: linearization (first order Taylor series
approximation)
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
Replication Methods
Use jackknife, bootstrap, or BRR
Enlarge the data set with additional columns of replication
weights
Work the same way for linear or nonlinear statistics
Preserve confidentiality (by not releasing certain design
information)
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
Issues with Replication Weights
Validity
–
–
–
–
Asymptotic Unbiasedness
Jackknife method is valid for restricted scenarios
Bootstrap method needs to be modified to fit into special designs
No replication method is valid for arbitrary sampling designs
Efficiency
– Small variance
– Replication variance estimators are often approximations to the
fully efficient V̂
– Efficiency often depends on sample sizes, number of replication
weights, and design features (such as sampling fractions)
Sparsity
– Want to reduce the size of replication.
– The number of jackknife replication weights depends on the
sample size
– Bootstrap methods typically require at least 1000 sets of weights
(StatCan uses 500!)
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
An Algebraic Construction of Replication Weights
Two research problems in this talk
1
How to construct a valid replication variance estimator ?
2
How to reduce the replication size without sacrificing the
efficiency (for some key items) ?
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
1
Introduction
2
Fully Efficient Replication Weights
3
Sparse and Efficient Replication Weights
4
Balanced Sampling
5
Simulation Studies
6
Concluding Remarks
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
Basic Settings
Consider an arbitrary sampling design with first and second order
inclusion probabilities πi and πij and basic design weights wi .
P
Develop variance estimator v(Ŷ) for Ŷ = i∈S wi yi .
The analytic variance estimator
XX
V̂ =
Ωij yi yj ,
i∈S j∈S
where Ωij = (πij − πi πj )/(πij πi πj ), is viewed as fully efficient.
Write V̂ = y0 ∆y, where ∆ = (Ωij ) is a n × n matrix and
y = (y1 , y2 , · · · , yn )0
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
An Algebraic Construction of Replication Weights
(k)
(k)
w(k) = (w1 , · · · , wn )0 , k = 1, · · · , L
P
(k)
Ŷ (k) = i∈S wi yi
Replication variance estimator
V̂R =
L
X
2
hk Ŷ (k) − Ŷ
(3)
k=1
for some known constants hk
Want to construct w(k) so that V̂R = V̂ for any y-variable.
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
Construction of Weights (Cont’d)
Approach 1: Fay’s Idea
Decomposition of ∆:
∆=
p
X
λk δ k δ 0k
(4)
k=1
where p = rank(∆) ≤ n
V̂ = V̂R for any y-variable if L = p and
w(k) = w + (λk /hk )1/2 δ k ,
where w = (w1 , · · · , wn )0 is the set of basic design weights.
Algebraically equivalent because
hk (Ŷ (k) − Ŷ)2 = y0 {hk (w(k) − w)(w(k) − w)0 }y
= y0 {λk δ k δ 0k }y
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
Construction of Weights (Cont’d)
Approach 2: Direct construction for some designs
In many single-stage designs, we can write V̂ = y0 ∆y where
∆ = ∆0 − ∆0 Z Z 0 ∆0 Z
−1
Z 0 ∆0
(5)
where ∆0 = diag{λ1 , · · · , λn }, λi ≥ 0 for all i = 1, 2, · · · , n and
Z 0 = (z1 , · · · , zn ) is a q × n matrix with full row rank.
After some matrix algebra, we can get
y0 ∆y = (y − Z β̂)0 ∆0 (y − Z β̂)
n
X
=
λk (yk − z0k β̂)2
k=1
where β̂ = (Z 0 ∆0 Z)−1 Z 0 ∆0 y.
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
Construction of Weights (Cont’d)
Deville(1999)’s approximation for variance estimation under
unequal probability of fixed-size sampling:
2
X
yi
− Ỹ
V̂ = c
(1 − πi )
πi
i∈S
−1
P
P
where c = 1 − i∈S b2i
, bi = (1 − πi )/ k∈S (1 − πk ) and
P
Ỹ = k∈S bi (yi /πi ).
Deville’s approximation uses λi = cπi−2 (1 − πi ) and zi = πi .
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
Construction of Weights (Cont’d)
Bredit and Chauvet
for balanced sampling
P (2011) approximation
P
with constraint i∈S xi /πi = i∈U xi :
V̂ =
n X
(1 − πi ) πi−2 (yi − ŷi )2
n−q
i∈S
where ŷi = x0i B̂ and
nP
o−1 P
−2
−2
0
B̂ =
(1
−
π
)π
x
x
k k
k k
k∈S
k∈S (1 − πk )πk xk yi .
Bredit and Chauvet (2011) approximation uses
λi = cπi−2 (1 − πi ) and zi = xi , where c = n/(n − q).
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
Jackknife for the designs with ∆ in (5).
2
Pn
0 β̂
1 Recall that we can write V̂ =
λ
y
−
z
k
k=1 k
k
P
(k)
n
2 Thus, we want to construct Ŷ (k) =
i=1 wi yi such that
2
2
hk Ŷ (k) − Ŷ
= λk yk − z0k β̂
which is achieved when
n
X
i=1
3
(k)
w i yi
=
n
X
i=1
wi yi − yk −
z0k β̂
λ 1/2
k
hk
Jackknife-type replication weights

n
o
 wi + (λk /hk )1/2 −1 + z0 (Z 0 ∆0 Z)−1 zi λi
k
(k)
o
n
wi =
1/2
0
0
 wi + (λk /hk )
zk (Z ∆0 Z)−1 zi λi
.
if i = k
otherwise.
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
Delete-a-Group Jackknife construction
1 Partition S into L groups: S = S ∪ S ∪ · · · S .
L
1
2
2 We can write
L X
2
2 X
X 0
0
λj yj − zj β̂
V̂ =
λj yj − zj β̂ =
k=1 j∈Sk
j∈S
3
P
(k)
Thus, we want to construct Ŷ (k) = ni=1 wi yi such that
2
2
X hk Ŷ (k) − Ŷ
=
λj yj − z0j β̂
j∈Sk
4
which is not always possible.
P
(k)
Instead, construct Ŷ (k) = ni=1 wi yi such that

2
X
2

1/2
(k)
hk Ŷ (k) − Ŷ
=
λj (2δj − 1) yj − z0j β̂


j∈Sk
(k)
where δj
are IID from Bernoulli(0.5).
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
Delete-a-Group Jackknife construction (Cont’d)
Note that


2
X 1/2 (k)
X E∗ {
λj (2δj − 1)(yj − z0j β̂)}2  =
λj yj − z0j β̂
j∈Sk
j∈Sk
(k)
where the expectation is with respect to the distribution for δi .
Delete-a-group Jackknife replication weights

o
n
P
0 (Z 0 ∆ Z)−1 z λ

q
z
if i ∈ Sk
w
+
−q
+

i i
0
ik
j∈Sk jk j
 i
(k)
wi =
o
n


 wi + P
otherwise.
qjk z0 (Z 0 ∆0 Z)−1 zi λi
j∈Sk
j
(k)
where qjk = (λj /hk )1/2 (2δj − 1).
Thus, it is essentially the same as applying BRR within each
random group.
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
1
Introduction
2
Fully Efficient Replication Weights
3
Sparse and Efficient Replication Weights
4
Balanced Sampling
5
Simulation Studies
6
Concluding Remarks
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
Efficiency and Sparsity
An initial L sets of replication weights w(k)
Weights from bootstrap, jackknife or BRR
Weights from the algebraic construction
V̂R is (assumed to be) fully efficient
L is very large
Goal: Reduce L to a much smaller number L0 without major loss
of efficiency
Strategy: Random sampling (random grouping) plus weight
calibration
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
Random Sampling of Weights
View V̂R =
PL
k=1
ck Ŷ (k) − Ŷ
2
as a population total
w(kj ) ,
Select
j = 1, · · · , L0 from w(k) , k = 1, · · · , L by simple
random sampling without replacement
2
P 0
(1)
Use V̂R = Lj=1
hkj (L/L0 ) Ŷ (kj ) − Ŷ
(1)
V̂R
is still a valid variance estimator:
(1)
E∗ (V̂R ) = V̂R
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
Random Sampling of Weights (Cont’d)
Under the algebraic construction of replication weights:
V̂R =
L
X
λk (δ 0k y)2
k=1
V̂R is a population total and λk can be viewed as a size variable
Select w(k) with inclusion probability ηk and use
(2)
V̂R
=
L0
X
hkj j=1
(2)
V̂R
ηkj
Ŷ (kj ) − Ŷ
2
(1)
is still valid and is more efficient than V̂R
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
Random Sampling of Weights (Cont’d)
Random selection of L0 sets of weights ensures validity of
replication variance estimator using the smaller number sets of
weights
(1)
None of V̂R
(2)
or V̂R
is as efficient as V̂R
Calibration over the selected smaller number sets of weights can
improve efficiency
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
Weight Calibration
Step 1
Collect all key variables z = (z1 , · · · , zm )0 from the survey
Compute the fully efficient variance-covariance matrix
L
X
(k)
0
(k)
V̂1 Ẑ =
hk1 Ẑ − Ẑ Ẑ − Ẑ
k=1
using the initial large number L sets of replication weights, where
(k)
Ẑ
=
X
(k)
wi zi
i∈S
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
Weight Calibration (Cont’d)
Step 2
Construct an initial set of L0 replication weights such that the
resulting variance estimator is also asymptotically unbiased.
(k)
(k)
(k)
Let w0 = (w10 , · · · , wn0 )0 , k = 1, · · · , L0
Less efficient variance estimator for Ẑ:
V̂0 (Ẑ) =
L0
X
(k)
(k)
hk0 (Ẑ0 − Ẑ)(Ẑ0 − Ẑ)0
k=1
(k)
where Ẑ0 =
P
i∈S
(k)
wi0 zi
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
Weight Calibration (Cont’d)
Step 3
Find T̂
(k)
=
P
∗(k)
i∈S
L0
X
wi0 zi such that
hk0 (T̂
(k)
− Ẑ)(T̂
(k)
− Ẑ)0 = V̂1 Ẑ .
(6)
k=1
To find T̂
1
2
(k)
satisfying (6), we use the following steps:
Pp0
Decompose V̂1 Ẑ into V̂1 Ẑ = k=1
αk qk q0k for some
αk > 0 and m-dimensional vector qk , k = 1, · · · , p0 . (p0 < m)
Construct
(k)
Ẑ + (αk /hk0 )1/2 qk , if k = 1, · · · , p0
T̂ =
Ẑ,
if k = p0 + 1, · · · , L0
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
Weight Calibration (Cont’d)
Step 4
Find
∗(k)
w0
∗(k)
∗(k)
= (w10 , · · · , wn0 )0 , k = 1, · · · , L0
that minimizes
X (k)
∗(k)
(k)
∗(k)
(k) 2
/wi0
D w0 , w0 =
τi wi0 − wi0
i∈S
subject to
X
∗(k)
wi0 zi = T̂
(k)
.
i∈S
P
∗(k)
The resulting replicate Ŷ ∗(k) = i∈S wi0 yi can take the form
(k)
(k)
(k) 0 (k)
Ŷ0 = Ŷ0 + T̂ (k) − Ẑ0
β̂
where β̂
(k)
=
nP
(k)
0
i∈S wi0 zi zi /τi
o−1 P
(k)
i∈S wi0 zi yi /τi .
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
Weight Calibration (Cont’d)
Step 4 (Cont’d)
The choice of τi = w−1
i is proposed (to make Cov(ê, Ẑ) = 0). In
this case,
V̂0∗
=
.
=
L0
X
k=1
L0
X
2
∗(k)
ck0 Ŷ0 − Ŷ
n
(k)
0 o2
(k)
ck0 ê0 − ê0 + T̂ − Ẑ β̂
k=1
0
.
= V̂0 ê + β̂ V̂1 Ẑ β̂ .
(7)
The first component is not fully efficient but the second
component is fully efficient.
In practice, negative replication weights need to be avoided.
∗(k)
(k)
Need to change the objective function D(w0 , w0 ) in the
calibration weighting for replication.
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
Properties of the Calibrated Weights
V̂0 (Ŷ): Less efficient variance estimator based on L0 replicates.
V̂1 (Ŷ): Fully efficient variance estimator based on L replicates.
(L >> L0 )
V̂0∗ (Ŷ): proposed calibration variance estimator based on L0
replicates.
Results
1
2
V{V̂1 (Ŷ)} ≤ V{V̂0∗ (Ŷ)} ≤ V{V̂0 (Ŷ)}.
Pm
.
If Ŷ = i=1 ai Ẑi , then V{V̂0∗ (Ŷ)} = V{V̂1 (Ŷ)}.
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
1
Introduction
2
Fully Efficient Replication Weights
3
Sparse and Efficient Replication Weights
4
Balanced Sampling
5
Simulation Studies
6
Concluding Remarks
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
Balanced sampling
Sampling design satisfying
X 1
X
xi =
xi .
πi
i∈S
(8)
i∈U
We are P
interested in creating replication variance method for
ŶHT = i∈S πi−1 yi .
P
Write ŶHT = i∈S πi−1 yi as
0
Ŷreg = ŶHT + X − X̂HT B̂
for some B̂.
Note that Ŷreg = ŶHT , regardless of B̂, under (8).
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
Balanced sampling
The choice of
(
)−1
X
X
B̂ =
πi−2 (1 − πi )xi x0i
πi−2 (1 − πi )xi yi
i∈S
i∈S
leads to
.
V Ŷreg = E
(
X
)
πi−2 (1 − πi ) (yi − ŷi )2
i∈S
where ŷi = x0i B̂.
Bredit and Chauvet (2011) approximation
n X
V̂ =
(1 − πi ) πi−2 (yi − ŷi )2 .
n−q
i∈S
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
Balanced sampling
Jackknife variance estimator
V̂JK =
n
X
2
∗(k)
hk ŶHT − ŶHT ,
(9)
k=1
∗(k)
(k)
(k)
where ŶHT = ŶHT + (X̂ − X̂HT )0 B̂,
P
(k)
(k)
(X̂HT , ŶHT ) = i∈S (k) πi−1 (x0i , yi ), hk = (1 − πk )n/(n − q), and
S (k) = S ∩ {k}c .
Note that
∗(k)
ŶHT − ŶHT
(k) 0
(k)
ŶHT − ŶHT + X̂ − X̂HT B̂
= −πk−1 yk − x0k B̂ .
=
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
1
Introduction
2
Fully Efficient Replication Weights
3
Sparse and Efficient Replication Weights
4
Balanced Sampling
5
Simulation Studies
6
Concluding Remarks
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
Settings
Family Expenditure Survey; N=2246; treated as a population
x1 : # Children; x2 : # Youth; x3 : # people; x4 : annual income; y1 :
total annual expenditure; y2 : expenditure on clothing; y3 :
expenditure on furnishings and equipment
Rao-Sampford PPS sampling method; known πi and πij
P
θ1 = Y1 = Ni=1 yi1 and
P
P
θ2 = Ȳ2 /Ȳ3 = N −1 Ni=1 yi2 / N −1 Ni=1 yi3
zi = (xi1 , xi2 , xi3 , xi4 , yi2 , yi3 )0
Relative bias and relative efficiency, based on 2000 runs
B
RB =
1 X V̂(b) − V
B
V
b=1
and
RE =
{MSE(V̂L )}1/2
,
{MSE(V̂)}1/2
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
Relative Bias (%) of Replication Variance Estimators for
θ = Y1
L0
n
25
50
50
V̂L
(1)
(2)
(3)
(4)
(C)
V̂R
V̂R
V̂R
V̂R
V̂R
−7.39
−7.74
−7.58
−6.48
−6.00
0.92
100
2.49
1.39
−0.63
1.80
2.95
2.13
150
−1.16
−1.09
−1.65
−1.96
2.96
0.72
100
1.73
−1.39
−2.02
0.58
2.91
−0.07
150
−1.16
1.62
0.04
−1.93
−0.74
−1.63
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
Relative Bias (%) of Replication Variance Estimators for
θ = Ȳ2 /Ȳ3
L0
n
25
50
50
V̂L
(1)
(2)
(3)
(4)
(C)
V̂R
V̂R
V̂R
V̂R
V̂R
−4.01
−3.45
−1.74
−5.37
−5.56
−2.20
100
−1.06
−0.39
3.37
0.39
−0.96
0.00
150
−0.12
−3.08
0.83
−7.16
−5.59
0.48
100
−1.18
−0.83
0.60
−2.12
−1.02
−1.31
150
−0.12
0.19
1.79
1.53
0.31
1.03
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
Relative Efficiency of Replication Variance Estimators
for θ = Y1
L0
n
25
50
50
V̂L
(1)
(2)
(3)
(4)
(C)
V̂R
V̂R
V̂R
V̂R
V̂R
1.00
0.84
1.05
0.89
0.93
1.34
100
1.00
0.55
0.80
0.92
0.93
1.34
150
1.00
0.47
0.70
0.83
0.76
1.00
100
1.00
0.83
1.00
0.97
0.98
1.61
150
1.00
0.57
0.72
0.95
0.96
1.28
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
Relative Efficiency of Replication Variance Estimators
for θ = Ȳ2 /Ȳ3
L0
n
25
50
50
V̂L
(1)
(2)
(3)
(4)
(C)
V̂R
V̂R
V̂R
V̂R
V̂R
1.00
0.78
0.80
0.69
0.74
0.98
100
1.00
0.54
0.57
0.35
0.49
1.00
150
1.00
0.50
0.52
0.42
0.55
0.99
100
1.00
0.74
0.77
0.60
0.64
0.99
150
1.00
0.60
0.61
0.35
0.50
0.99
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
1
Introduction
2
Fully Efficient Replication Weights
3
Sparse and Efficient Replication Weights
4
Balanced Sampling
5
Simulation Studies
6
Concluding Remarks
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
Remarks
Construction of sparse and efficient replication weights is a task
at the data file preparation stage
Issues with computational implementation need to be dealt with
by those who produce public-use data files, not the data users
The algebraic construction of fully efficient replication weights
can serve as a starting point
The proposed strategy, i.e. random selection plus weight
calibration, on producing sparse and efficient replication weights
can be applied to any initial weights from jackknife, bootstrap or
BRR methods
Identify and include all key variables at the calibration stage
To produce a small number (say 30-50) sets of replication
weights which provide valid and efficient variance estimation is
an important research problem with both theoretical and
practical significance
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Introduction Fully Efficient Replication Weights Sparse and Efficient Replication Weights Balanced Sampling Simulation Studies Concluding
Dedication
This work is dedicated to the memory of
Professor Randy Sitter
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