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Estimation of Sampling Errors, CV, Confidence Intervals . Arun Srivastava. Properties of a good Estimator. Unbiasedness Efficiency Variance measures precision of an estimator Mean square error measures it’s accuracy Consistency Concept of Bias - PowerPoint PPT Presentation
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Arun Srivastava
Properties of a good EstimatorUnbiasednessEfficiency
Variance measures precision of an estimator Mean square error measures it’s accuracy
ConsistencyConcept of BiasWhy estimation of sampling error is so
important?
Simple random sampling (SRS): Sample mean is an unbiased estimator of
population mean.
; SRSWR
SRSWOR
nsyV2
)(ˆ
n
ii yy
ns
1
22 )(11
211)(ˆ sNn
yV
Systematic SamplingAn approximate estimator of variance is
If population is assumed to be in random order
1
1
211
11121)(ˆ
n
iii yy
nNnyV
211)(ˆ sNn
yV
PPSWR Sampling
n
i i
ipps p
yn
Y1
1ˆ
2
1
ˆ)1(
1)ˆ(ˆ
n
ipps
i
ipps Y
py
nnYV
Varying probability sampling (without replacement):Horvitz –Thompson estimator
For IPPS
n
i i
iHT
yY
1
ˆ
n
i i
iHT p
yn
Y1
1ˆ
Stratified samplingEstimator of total and estimated variance are
h
L
hhst yWY
1
ˆ
2
1
2 11)ˆ(ˆ h
L
h hhhst s
NnWYV
Cluster sampling
Estimator of mean and variances are
n
iicl y
nY
1
1ˆ
2
1
)(1
1)11()ˆ(
N
iicl YY
NNnYV
Cluster Sampling (Contd.)Estimator of variance
Variance formula is also given by
2
1
)ˆ(11)11()ˆ(ˆ
n
iclicl Yy
nNnYV
)1(1)ˆ(2
MnSYV cl
Cluster Sampling (Contd.)Intra-class correlation
Intra-class correlation is the correlation coefficient between pair of units that are in the same cluster. It measures intra-cluster variability.
2
22
)1( SMSSb
Multi-stage SamplingEstimator of total
Variancei
n
ii
m
jij
n
i i
its yM
nNy
mM
nNY
i
111
ˆ
2
1
222 1111ˆi
ii
N
iibtts S
MmM
nNS
NnNYV
Multi-stage Sampling (Contd.)Estimator of variance
In case of equal clusters
2
1
222 1111ˆˆi
ii
n
iibts s
MmM
nNs
NnNYV
n
ii
n
i
m
jij y
ny
nmY
11 1
11ˆ
22 11111ˆwb S
MmnS
NnYV
Multi-stage Sampling (Contd.)Estimator of
variance
nsb2
22 11111ˆˆwb s
MmNs
NnYV
Sample weights Base weights Non response adjustments Post-stratification adjustments Base weights are inverse of selection
probabilities Weights provided to ultimate sampling
units
Sample weights (Contd.)For unequal probability wor sampling
For two-stage sampling with pps systematic selection at the first stage and equal probability selection at the second stage weights are (define notations)
i
n
ii
n
i i
iHT yw
yY
11
ˆ
i
i
nmXXM