1) The document discusses probability proportional to size (PPS) sampling techniques, including PPS sampling with and without replacement.
2) Cumulative total and Lahiri's methods are described for PPS sampling with replacement, while general selection procedure, Sen-Midzuno method, and Narain's scheme are covered for PPS sampling without replacement.
3) Estimators for population totals, means, variances, and inclusion probabilities are defined for PPS samples. Both ordered and unordered estimators are discussed.
3. Outline…
Background study of PPS sampling or Why
PPS sampling (re45ppview of SRS, Stratified,
Systematic etc)
What is PPS sampling
Sampling unit selection procedure
Estimators (ordered & unordered)
4. Simple Random Sampling (SRS)
Population
Sample
30
16
1. Homogeneous
2. Equal probability
3. Simple in concept
5. Simple Random Sampling (SRS)
Procedure of selecting a random sample
Lottery method
Use of random number table
Remind basic concept of estimating mean,
total, variance and their properties.
Different schemes of using random number
table.
7. Stratified sampling
• Heterogeneous
• Do SRS in each stratum
• Calculate mean, total, variance or measuring
statistics for each strata and combine them.
• Study the allocation rules
Equal allocation
Proportional allocation
Neyman allocation
Optimum allocation
• Gain in precision
9. Systematic Sampling
• Sample selection procedure
Linear systemic sampling
Circular systematic
• Estimate total, mean and variance
• Study their properties
• Gain in precision
13. Probability proportional to size(PPS)
• Procedures of selecting a sample with
replacement
Cumulative total method
Lahiri’s method
• Procedures of selecting a sample without
replacement
General selection procedure
Sen-midzuno method
Narain’s scheme of sample selection
14. Cumulative total method (PPSwr)…
• Sampling procedure
S.N. of
holdings
Size
(Xi)
Cumulative
size
Numbers
associated
1 50 50 1-50
2 30 80 51-80
3 45 125 81-125
4 25 150 126-150
5 40 190 151-190
6 26 216 191-216
7 44 260 217-260
8 35 295 261-295
9 28 323 296-323
10 27 350 324-350
1. Random number less
than equal max
cumulative size (350).
2. Let it 272, it lies
between 261-295. 8th
holding is selected.
3. 346, 165 and 044
random number thus
10th , 5th and 1st holding
selected.
4. 8th , 10th , 5th and 1st
unit makes sample
15. Cumulative total method (PPSwr)…
• Drawback : This procedure involves writing
down the successive cumulative totals. This is
time consuming and tedious if the number of
units in the population is large.
16. Lahir’s Method (PPSwr)
N = Number of units; M=Maximum size units
= Size of k th unit
1 - N
l1 - M
k
kX
Select a
random
number
Accept k th unit if (k, l < )
Reject k th unit if (k, l > )
kX
kX
17. Lahir’s Method(1951)
• Referring to the random number table, the pair
is (10, 13). Hence the 10 th unit is selected in
the sample.
• Similarly, choosing other pairs, we can have
(4, 26), (5, 35), (7,26). (4, 26) rejected. Why???
• Another pair (8, 16) .
• Sample is 10, 5, 7 and 8 th unit
18. Lahir’s Method(1951)
• Advantage:
– It does not require writing down all cumulative
totals for each unit.
– Sizes of all the units need not be known before
hand. We need only some number greater than the
maximum size and the sizes of those units which
are selected by the choice of the first set of random
numbers 1 to N for drawing sample under this
scheme.
19. Lahir’s Method(1951)
• Disadvantage:
– It results in the wastage of time and efforts if units
get rejected. The probability of rejection 1 −
𝑋
𝑀
.
• The expected numbers of draws required to draw one
unit
𝑀
𝑋
.
• This number is large if 𝑀 is much larger than 𝑋
20. Journey: Sample to Population
• Total, Mean, Variance
Sample
Sample
mean
Sample
variance
Sample
total
Population
Population
mean
Population
variance
Population
total
• Sample total/mean with unbiased/biased
estimator pop total/mean having population
variance.
• Sample variance unbiased/biased estimator of
pop variance
21. Journey: Sample to Population
𝐸 𝑡 /𝑚 = 𝑇 /𝑀 with variance 𝑉.
and 𝐸 𝑣 = 𝑉
𝐸 𝑡 /𝑚 = 𝑇 /𝑀 with variance 𝑣
Estimators Sample Population
Total 𝑡 𝑇
Mean 𝑚 𝑀
Variance 𝑣 𝑉
22. Expectation
Sample-1 Sample-2 Sample-3 Sample-k
𝑠1 𝑠2 𝑠3 𝑠 𝑘
𝐸𝑥𝑝𝑒𝑐𝑡𝑎𝑡𝑖𝑜𝑛 𝐸[𝑠] =
1
𝑘
𝑖=1
𝑘
𝑠𝑖
𝑠𝑖 be any
statistic like
mean
variance,
standard
deviation
… … …
… … …
𝑃𝑜𝑝 𝑛
𝑠𝑖𝑧𝑒 = 𝑁
Sample size=n
Sample
𝑁
𝑛
= 𝑘
23. IID random variables
𝑥1 𝑥2 𝑥 𝑛𝑥3
𝐷𝑖𝑠𝑡 𝑛 𝐷𝑖𝑠𝑡 𝑛 𝐷𝑖𝑠𝑡 𝑛 𝐷𝑖𝑠𝑡 𝑛
Look like twin but not.
They comes from different
mother.
IID
random
variables
… … …
24. Defining random variale…
• 𝑦𝑖 = Value of the characreristics under study
(𝑦𝑖 ?? ambiguity?? Next slide)
• 𝑁 = Population size
• 𝑝𝑖 = 𝑋𝑖/𝑋
• 𝑧𝑖 =
𝑦 𝑖
𝑝 𝑖
; 𝑖 = 1, 2, 3, … , 𝑛 IID
random variable....... Why ????
• 𝑝𝑖 =
1
𝑁
Simple Random Sampling
25. Example 5.3
• Selected sample (cummulative/Lahiri’s method)
Area under
Crop
5.2 5.9 3.9 4.2 4.7 4.8 4.9 6.8 4.7 5.7
Yield of crop 28 29 30 22 24 25 28 37 26 32
Area under
Crop
5.2 5.2 4.9 4.0 1.3 7.4 7.4 4.8 6.2 6.2
Yield of crop 25 38 31 16 6 61 61 29 47 47
Size (X)
Value of the
characteristic under
study (Y)
(N=100, n=20)
26. Estimators…
Theorem 5.3.1: In pps sampling, wr, an unbiased
estimator of the population total Y is given by
𝑌𝑝𝑝𝑠 =
1
𝑛
1
𝑛
(𝑦𝑖/𝑝𝑖)
With its sampling variance
𝑉( 𝑌𝑝𝑝𝑠) =
1
𝑛 1
𝑁
𝑝𝑖 (
𝑦 𝑖
𝑝 𝑖
− 𝑌)2
*** find unbiased estimator of mean....
***See corollary
27. Estimators…
• Theorem: In pps sampling, wr, an unbiased
estimator of 𝑉( 𝑌𝑝𝑝𝑠)is given by
• 𝑣( 𝑌𝑝𝑝𝑠) =
1
𝑛(𝑛−1) 1
𝑛
(
𝑦 𝑖
𝑝 𝑖
− 𝑌𝑝𝑝𝑠)2
=
1
𝑛(𝑛−1)
[ 1
𝑛
(
𝑦 𝑖
𝑝 𝑖
)2
−𝑛 𝑌𝑝𝑝𝑠
2
]
28. Gain due to pps sampling...
• Study gain due to PPS sampling with
replacement
33. PPS sampling WoR
• General selection procedure 𝑝𝑖 = 𝑋𝑖/𝑋
Select a pair of random numbers 𝑖, 𝑗 𝑠. 𝑡. ( 𝑖 ≤
Orchard 1 2 3 4 5 6 7 8
Trees 50 30 25 40 26 44 20 35
Orchard 1 2 3 4 Blank 5 6 7
Trees 50 30 25 40 44 20 35
34. PPS sampling without replaceent…
• The first order incluson probability for unit 𝑖 is
the probability that 𝑖 is included in a sample of
size n and is given by
𝜋𝑖 = 𝑠∋𝑖 𝑝(𝑠) .
• The second order inclusion probability for unit 𝑖
and 𝑗 is defined as the probability that the two
units 𝑖 and 𝑗 are included in a sample of size n
𝜋𝑖𝑗 = 𝑠∋𝑖,𝑗 𝑝(𝑠) .
39. Ordered and unordered estimator
• Ordred estimator: Incorporate sampling unit’s
order. Need only conditional probability not
inclusion probability.
• Unordered estimator: Free from order concept
of sampling unit’s ordes. Incorporate inclusion
probability.
40. Ordered and unordered estimator
• Das-Raj’s ordered estimator→ No need
inclusion probability
• Horvitz-Thompson estimator
(H-T estimator)
• Murthy’s estimator
Unordered
estimator need
inclusion
probability