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