Ratio Estimation - Survey Sampling Techniques - Lecture Slides, Slides of Survey Sampling Techniques

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CSE

ratio

estimation

for

population

mean

 

 

 

 

n i

i

n i

i

i

n i

i

n i

i

r

M

y

M

t M

y

1 1

(^11)

U

y

CSE

ratio

estimation

for

population

mean

^



^

^

^

















^ 

^  

n i

i

i

U

n i

r

i i

n i

r i

i i

r

n i

i i

i i

i

r

U

r

M

n

M

M

M

y y M n y M y M n s

s m

m M

M

N

n

s n

n N

M

y V

1

S

1

2

2

1

2

2

1

2

2

2

2

or

of

mean

sample

by

estimated

be

can

docsity.com

Dorm

example

   n i

i

n i

i

r^

t M

y

(^11)

^

^

 n i

r

i

i

r

y

y

M

n

s

1

2

2

2

Dorm

example

^

^

^  

^  

n i

i i

i i

i

r

r

s m

m M

M

N

n

s n

n N

M

y

V

1

2

2

2

2

Dorm

example





r

r

y

K

t

ˆ

ˆ













r

r

y

V

K

t

V

ˆ

ˆ

ˆ

ˆ

Coots

egg

example

Target

pop

=

American

coot

eggs

in

Minnedosa,

Manitoba

PSU

/

cluster

=

clutch

(nest)

SSU

/

element

=

egg

w/in

clutch

Stage

1

SRS

of

n

clutches

N

clutches,

but

probably

pretty

large

Stage

2

SRS

of

m

= i

from

M

i^

eggs

in

a

clutch

Do

not

know

K

eggs

in

population,

also

large

Can

count

M

= i^

eggs

in

sampled

clutch

i

Measurement

y ij^

volume

of

egg

j

from

clutch

i

Coots

egg

example

Estimation

goal

Estimate

,^

population

mean

volume

per

coot

egg

in

Minnedosa,

Manitoba

What

estimator?

Unbiased

estimation

•^

Don’t

know

N

total

number

of

clutches

or

K

total

number

of

eggs

in

Minnedosa,

Manitoba

Ratio

estimation

•^

Only

requires

knowledge

of

M

, i^

number

of

eggs

in

clutch

i

in

addition

to

data

collected

U

y

Coots

egg

example

Clutch

M

i

y^ i

(^2) i s

ˆti

i i i i

s m M M

2 2

2 1



^ 



2 ˆ

ˆ^



^ 



r i

i

y M t

1

13

0.00 94

50.235 94

0 .67190 1

2

13

0.00 09

54.524 38

3

6

0.00 05

5.497 50

0 .00577 7

4

11

0.00 08

0 .03935 4

5

10

0.00 02

24.957 08

0 .00629 8

0.0 02631

6

13

0.00 03

51.795 37

0 .02362 2

7

9

0.00 51

17.343 62

0 .15944 1

8

11

0.00 51

32.576 79

0 .25358 9

9

12

0.00 01

41.526 95

0 .00639 6

10

11

0.02 24

32.576 79

1.10866 4

…^

…^

…^

…^

…^

…^

…

18 0

9

0.00 01

17.519 18

0 .00239 1

18 1

12

0.00 17

41.439 34

0 .10233 9

18 2

13

0.000 03

54.858 54

0 .00262 5

18 3

13

0.00 88

57.392 62

0 .63056 3

18 4

12

0.0000 06

0 .00040 0

sum

1757

4375.9 47

4 2.1744 5

11,43 9.

var

149.5 65814

 ˆ^ y^ r

CSE2:

Unbiased

vs.

ratio

estimation

Unbiased

estimator

can

have

poor

precision

if

Cluster

sizes

(

M

i^ )^

are

unequal

t^ i

(cluster

total)

is

roughly

proportional

to

M

i^

(cluster

size)

Biased

(ratio

estimator)

can

be

precise

if

t^ i

roughly

proportional

to

M

i

This

happens

frequently

in

pops

w/cluster

sizes

(

M

) i

vary

Inclusion

probability

for

an

element

under

2

‐ stage

cluster

sampling

using

SRSWOR

at

each

stage

(CSE2)



i^

P{cluster

i

in

sample}

=

n

/

N



j^ |

i^

Pr

{element

j

given

cluster

i

in

sample}

=

m

i^ /

M

i



ij^

=

Pr

{element

j

and

cluster

i

in

sample}

=



i 

j^ |

i

=

(

n

/

N

)

x

(

m

/ i

M

) i^

=

nm

i^ / NM

i

CSE2:

Self

‐

weighting

design

Stage

1:

Select

n

PSUs

from

N

PSUs

in

pop

using

SRS

Stage

2:

Choose

m

i^

proportional

to

M

i^

so

that

m

i^ /

M

i^

is

constant,

use

SRS

to

select

sample

Sample

weight

for

SSU

j

in

cluster

i

is

constant

for

all

elements

Weight may vary slightly in practice because may not bepossible for

m

/i^

M

to be equal to 1/i

c^

for all clusters c N n

M m N n

w

i i

ij^

CSE2:

Self

‐

weighting

design

Are

dorm

student

or

coot

egg

samples

self

‐ weighting

2

‐ stage

cluster

samples?

Self

weighting

designs

SRS

SYS

STS

with

proportional

allocation

CSE

CSE

with

m

i^

proportional

to

M

i

or

c

=

M

/ i m

i

c N n

M m N n

w

i i

ij^

N^ n

w

ij

N^ n

N n

w

h h

hj

N^ n

w

i^

Ch

Unequal

probability

cluster

samples

Use

unequal

selection

probabilities

to

sample

clusters

to

save

costs

and

improve

precision

for

a

given

budget

Focus

is

on

PPSWR

as

stage

1

design

But

MANY

other

unequal

probability

designs

are

possible