Title Sampling Theory and Methods Mean Squared Error Estimator Bias Of An Estimator Sampling (Statistics) Errors And Residuals 3.8 MB 191
```                            Preface
Contents
1 Preliminaries
1.1 Basic Definitions
1.2 Estimation of Population Total
1.3 Problems and Solutions
2 Equal Probability Sampling
2.1 Simple Random Sampling
2.2 Estimation of Total
2.3 Problems and Solutions
3 Systematic Sampling Schemes
3.1 Introduction
3.2 Linear Systematic Sampling
3.3 Schemes for Populations with Linear Trend
3.4 Autocorrelated Populations
3.5 Estimation of Variance
3.6 Circular Systematic Sampling
3.7 Systematic Sampling in Two Dimensions
3.8 Problems and Solutions
4. Unequal Probability Sampling
4.1 PPSWR Sampling Method
4.2 PPSWOR Sampling Method
4.3 Random Group Method
4.4 Midzuno scheme
4.5 PPS Systematic Scheme
4.6 Problems and Solutions
5 Stratified Sampling
5.1 Introduction
5.2 Sample Size Allocation
5.3 Comparision with Other Schemes
5.4 Problems and Solutions
6 Use of Auxiliary Information
6.1 Introduction
6.2 Ratio Estimation
6.3 Unbiased Ratio Type Estimators
6.4 Almost Unbiased Ratio Estimators
6.5 Jackknife Ratio Estimator
6.6 Bound for Bias
6.7 Product Estimation
6.8 Two Phase Sampling
6.9 Use of Multi-auxiliary Information
6.10 Ratio Estimation in Stratified Sampling
6.11 Problems and Solutions
7 Regression Estimation
7.1 Introduction
7.2 Difference Estimation
7.3 Double Sampling in Difference Estimation
7.4 Multivariate Difference Estimator
7.5 Inference under Super Population Models
7.6 Problems and Solutions
8 Multistage Sampling
8.1 Introduction
8.2 Estimation under Cluster Sampling
8.3 Multistage Sampling
9 Non-sampling Errors
9.1 Incomplete Surveys
9.2 Randomised Response Methods
9.3 Observational Errors
10 Recent Developments
10.2 Estimation of Distribution Functi'ons
10.3 Random˜sed Response Methods for Quantitative Data
References
Index
```
##### Document Text Contents
Page 2

Sampling Theory
and Methods

S. Sampatb

CRC Press
Boca Raton London New York Washington, D.C.

<iJ
Narosa Publishing House
New Delhi Chennai Mumbai Calcutta

Page 95

88 Sampling Theory and Methods

I

L(j)- L(j + n = . _; L<i + 1- 2u)pu
](} -1) u=l

(5.33)

If S stands for the summation term in the right hand side of (5.33), grouping
together the terms equidistant from the beginning and end, S can be written as

m

S = L[2m + 1- 2u][pu - P:!m+l-u 1 if j=2m is even
u=l
m

S = L [2m+ 2- 2u ][p u - P2m+2-u 1 if j=2m+ 1 is odd
u=l

Since P; ~ Pi+l for all i, every term in S is non-negative. Therefore S is non-

negative. Hence we conclude that L is a non-decreasing function. Therefore
L(nk) ~ L(k). This leads to the conclusion that the average variance of the

conventional estimator under simple random sampling is larger than the average

variance of the estimator Yo introduced in this section. However no such
general result can be proved about the efficiency of systematic sampling relative
to simple random sampling or stratified sampling unless further restrictions are
imposed on the correlations p u . The following theorem is due to Cochran
(1946).

Theorem 5.4 If Pi~ Pi+l ~O.i = 1. 2 .... , N -2 .al = Pi+2 + P; -2Pi+l ~0.
and i = l, 2, ... , N -2 then EM[V(YLSs )] ~ EM[V(fo)] S EM[V(~,rs)] .

., .. ..
Furthermore, unless () F = 0, i = 1, 2 •.... N- 3.£ M [V(Y LSS )] S EM [V (Y LSS )]

Proof As a? ~O.wehave Pi+2+P;-2Pi+l ~O.i=l.2, ... ,N-2
By induction it can be shown that Pi+c+l - Pi+c ~ Pi+c- Pi for any integer c.
Hence for any integer a,c>O we have

i+c+l a+c-1

LPi+c+l- Pi+c ~ L Pi+c- P;
i=a i=a

which gives Pa+2c + Pa- 2Pa-t-c ~ 0
Consider the difference

EM [V(Yo)]-EM [V(YLSs )] =

(5.34)

2a 2 (k l)N 2 [nk-1 n-1 k-1 ]
2 - L(nk-u)pu -k

2 L(n-U)Pku -nL(k-u)pu (5.35)
Nnk (k -1) u=l u=l u=l

nk-1 k n-1

Note that L (nk- u)pu = L L[nk- (i + jk)]P;+ jk
u=l i=l j=O

Page 96

Stratified Sampling 89

k-1 n-1 n-1

= LL[nk -(i+ jk)]P;-..Jk + L<n- j)p jk
•='I j=O j=l

k-1 n-1 n-1

= LL(n- j)(k-i)Pi+jk +k L,<n- j)p jk +
i=l j=l j=l

k-1 n-2 k-1

LLi<n- j-i>p jk+i +n L<k-i)p;
i=l j=O i=l

(5.36)
k-1 n-2 k-1 n-i

Since L L i(n- j- i)p jk+i = L L i(n- j) p jk-(k-i)
i=l j=O i=l j=l

k-1 n-i

= LL(k- i)(n- j)p jk-i (5.37)
i=l j=l

The expression inside square braces of (5.35) can be written as
k-1 n-i k-1 n-i n-1

LL(k -i)(n- j)p jk+i + LL(k- i)(n- j)p jk-i -k(k -l)L,p jk
i=l j=l i=l j=l j=l

k-1 n-i

which is equal to L L (k - i)(n- j)[p jk+i + P jk-i - 2p jk] ·
i=l j=l

By (5.34) this is clearly non-negative. Therefore EM [V(fo)] S EM [V(YLSs )].
Further from (5.38), it can be seen that the above inequality will be strict if and

only if (); = 0, i = 1, 2, ...• N -1. Hence the proof. •

5.4 Problems and Solutions

Nt
Problem 5.1 A sampler has two strata with relative sizes W1 =-and

N

N2
W 2 =- . He believes that S 1 • S 2 can be taken as equal. For a given cost

N
C = c1n1 + c2n 2 • show that (assuming N his large)

[
Vprop]= [W1c1 +W2c2]

Vopt [wl~ +W2~r

Solution When Nh is large, V(Ysr) = ±{N~ N h -nh }s~'
h=l N hnh

Page 190

Index
--------------------------------------------------

almost unbiased ratio type estimator,

104,105
autocorrelated populations, 39,87
auxiliary information, 97-121
balanced systematic sampling, 35-37
Bartholomew, 154
Bellhouse, 47
bias, 2
bound for bias, 105
centered systematic sampling, 34
Chambers, 171,173
Chaudhuri, 161
circular systematic sampling, 43,44
cluster sampling, 140
Cochran, 63,88
cost optimum allocation. 82
cumulative total method, 55
Dalenius, 154
Das, 47
Deming's model, 154
Desraj ordered estimator, 60
difference estimator, 124-126
distribution.function, 171
Dunstan, 171,173
edge unit, 167
El-Bardy, 152
entropy, 3
Erikkson, 174
finite population, I
Folsom's model, 160
Garg, 38
Gauss-Markov, 132
Hansen and Hurwitz, 152
Harltey, 63,70,102,106
Hess, 154
Horvitz-Thompson. 3,6,8,63
implied estimator, 129
inclusion indicators, 4
inclusion probabilities, 4,5

incomplete surveys, 152
Jindal, 38
Kish, 154
Kovar, 171,173
Kuk, 172
Kovar, 171,173
Kuk, 172
Kunte. 44
Lagrangian multipliers, 81,93
Lahiri, 43,56
linear systematic sampling, 29-32
Mantel, 171,173
mean squared error, 1 ,3
Midzuno, 67-70
model unbiasedness, 131
modifed Hansen-Hurwtiz

estimator, 168
modified Horvitz-Thompson

estimator, 170
modified systematic sampling,

38,39
multi-auxiliary information, 113
multistage sampling, 140-150
Murthy's unordered estimator, 62
neighbourhood, 167
network, 167
Neyman optimum allocation, 81
non-sampling errors, 152-164
observational errors, 161
Olkin, 113
parameter 1 ,3
Politz-Simmons technique, 156
population siie, 1
pps systematic scheme, 70
ppswor, 60
ppswr,55
probability sampling, 1
product estimation, 106-108
proportional allocation, 79

Page 191

l84 References

Quenouille. 47
random group method. 63
randomised response 15 8-161 . 17 4
Rao. 4.63.70,1 02.171,173
ratio estimator. 97-1 05
regression estimation. 122-124
Ross. 106
Royall, 137
sample size allocation. 79-85
sample, I
sampling design, 2,3,4,5
sampling in two dimensions. 44,45
Sarndal, 16
Sethi,35
Simmons. 159
Sethi. 35
Simmons. 159

simple random sampling, I 0-28
Singh,38
Srinath. 154
srswr. 25
statistic. 2
stratified sampling, 76-96,115
super-population model, 129
systematic sampling, 29-54
Thompson, 165.171
two phase sampling, 108-112
two stage sampling, 140-150
unbiased ratio type estimators, ! 00
unbiasedness, 2
unequal probability sampling, 55

·Warner, 158
Yates. 33.73
Yates-Grundy, 7