DOUBLE SAMPLING FOR RATIO ESTIMATOR
ALIFATHIMA ABSANA H
2048111
INTRODUCTION:
In
many human surveys, information is in the most cases are not obtained from all the
units in the survey, even after some callbacks. An estimate obtained from such
incomplete data may be misleading especially when the respondents differ from
the non-respondents because the estimate can be biased. Hansen and Hurwitz
(1946) proposed a technique for adjusting for non-response to address the bias
problem. Their idea is to take a subsample from the non-sample respondents to
get an estimate for the subpopulation represented by the non-respondents.
DOUBLE SAMPLING:
Double Sampling can also be referred to as
the Two-phase Sampling. Auxiliary information has always been seems effective
in increasing the precision of estimates in survey sampling in which the
precision of estimates of the mean of the variable of interest is increased by
the presence of highly correlated auxiliary variables. There are situations
when auxiliary information is available at the population level and the cost of
collecting the variable of interest per unit is affordable, then single-phase
sampling is more appropriate. But when prior information on auxiliary variable
is lacking, then it is neither practical nor economical to conduct a census for
this purpose. Therefore, an appropriate technique employed to get estimates of
auxiliary variables on the basis of samples is Double-phase sampling. This
technique is used when the cost of obtaining estimates of the variable of
interest directly from the population is expensive or impracticable. The theory
of Double sampling is presented under the assumption that one of the sample is
a subsample of the other. This type of sampling technique is called Double
Sampling Technique or Sampling followed by sub-sampling.
RATIO ESTIMATION WITH DOUBLE
SAMPLING:
- yi - variable of interest
- xi - auxiliary variable
- n' -
number of units in the first sample (which includes the second sample)
- n - number of units in the second
sample
Only in the second samples, both xi and yi values
are observed. In the remaining units, (in the first but not the second
sample), xi but not yi are observed. Note that
observing yi's are expensive whereas observing xi's are not.
If xi and yi are highly linearly correlated and
approximately passing through the origin, then the ratio estimate with double
sampling may lead to improved estimates. While using the ratio estimate for
double sampling, the ratio will be estimated using samples where both (x, y)
are observed, i.e., the second sample, whereas τx will be estimated
by the larger first sample.
The ratio estimator is:
ANALYSIS OF A REAL LIFE APPLICATON USING R:
A forest resource manager is
interested in estimating the total number of dead trees in a 400-acre area of
heavy infestation. She subdivides the area into 200 plots of equal sizes and
uses photo counts to find the number of dead trees in 18 randomly sampled
plots. She then randomly samples 8 plots out of these 18 plots and conducts a
ground count on these 8 plots. Compute the estimated variance of the ratio
estimator. the ratio estimate for the
population total.
Let x denote
the number of dead trees in the plot by photo count and y the
number of dead trees by ground count. The data are given as:
|
Plot |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
|
x' |
5 |
7 |
10 |
6 |
7 |
9 |
3 |
6 |
8 |
11 |
5 |
9 |
12 |
13 |
3 |
20 |
15 |
4 |
Out of these 18 plots, 8 are randomly selected and a ground count is
conducted.
|
Plot |
2 |
3 |
5 |
6 |
12 |
15 |
16 |
17 |
|
x |
7 |
10 |
7 |
9 |
9 |
3 |
20 |
15 |
|
y |
9 |
13 |
10 |
11 |
10 |
4 |
25 |
17 |
|
y-rx |
0.3375 |
0.6250 |
1.3375 |
-0.1375 |
-1.1375 |
0.2875 |
0.2500 |
-1.5625 |
|
|
|
|
|
|
|
|
|
|
APPLICATIONS:
Double sampling can be employed
for estimating the number of transmission sources in a Poisson process, for
biomass estimates, for correcting helicopter counts of moose, for estimating
sizes of animal populations, for estimation related to welfare programs, for
evaluating a new medical diagnostic test, for estimating employment rates, for
estimation in forestry. The double sampling method can be employed for
determining the sample size for estimation related to rare items
ADVANTAGES:
The potential advantage of using a double sampling plan is that it may reduce the average number of inspections, compared to the equivalent single sampling plan. It means, while the double sampling plan is as effective as the single sampling plan, the former is less expensive to conduct. From a psychological viewpoint, it is desirable to have a second chance Sometimes using a double sampling plan is the only way of chance. Sometimes using a double sampling plan is the only way of convincing upper management to implement a statistically valid sampling plan
DISADVANTAGES:
Unless
curtailment(curtailment refers to reject a
lot without complete inspection of the second sample)is used on the
second sample, under some circumstances double
sampling may require more total
inspection than would be required in a single-sampling plan that offers the
same protection. So, unless double sampling is used carefully,
its potential economic advantage may be lost. The second disadvantage of double sampling is
that it is administratively more complex, which may increase the opportunity
for the occurrence of inspection errors.
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