PPSWOR USING HT ESTIMATE AND COMPARING IT WITH YT ESTIMATE
PPSWOR USING HT ESTIMATE AND COMPARING IT WITH YT ESTIMATE
-DARSHAN BINAYKIA JAIN_2148104
PPS (PROBABILITY PROPORTIONAL TO SIZE)
Probability
proportional to size (PPS) sampling is a method of sampling from a finite
population in which a size measure is available for each population unit before
sampling and where the probability of selecting a unit is proportional to its
size. When the units vary in size
and the variable under study is highly correlated with the size of the unit,
the probability of selection maybe assigned in proportion to the size of the unit
and this type of sampling where the probability of sampling is proportional to
the size of the unit is known as Probability proportional to size.
EXAMPLE:
1. 1. In industrial surveys, the number of workers in the factory can be considered as the measure of size when studying the industrial output from the respective factory.
2. 2. If one sample had 20,000 members, the probability of a member being selected would be 1/20000 or .005 percent. If another sample had 10,000 members, the chance of a member being selected would be 1/10000 or .01 percent.
COMPARING SRS with PPS:
The difference between simple random sampling and PPS sampling is that the probability of drawing any specified unit at any given draw is the same in the former method. In the latter method, however, the probability varies from draw to draw. As a result, PPS sampling theory is more complicated than simple random sampling theory. It appears in pps sampling that such procedure would give biased estimators as the larger units are overrepresented and the smaller units are under-represented in the sample. This will happen in the case of the sample mean as an estimator of the population mean where all the units are given equal weight. Instead of giving equal weights to all the units, if the sample observations are suitably weighted at the estimation stage by taking the probabilities of selection into account, then it is possible to obtain unbiased estimators.
In pps sampling, there are two possibilities to draw the sample, i.e., with replacement and without replacement.
METHODS OF PPS SAMPLING:
SELECTION OF UNITS WITH REPLACEMENT: The probability of selection of a unit will not change, and the probability of selecting a specified unit is the same at any stage. There is no redistribution of the probabilities after a draw.
SELECTION OF UNITS
WITHOUT REPLACEMENT: The probability of selection of a unit will change at any
stage, and the probabilities are redistributed after each draw.
VARYING PROBABILITY SCHEME WITHOUT REPLACEMENT:
In varying probability scheme without replacement, when the
initial probabilities of selection are unequal, then the probability of drawing
a specified unit of the population at a given draw changes with the draw. Generally,
the sampling WOR provides a more efficient estimator than sampling WR. The
estimators for population mean and variance are more complicated. So, this
scheme is not commonly used in practice, especially in large scale sample
surveys with small sampling fractions.
NOTATION:
ORDERED ESTIMATOR:
To overcome the difficulty of changing expectation with each
draw, associate a new variate with each draw such that its expectation is equal
to the population value of the variate under study. Such estimators take into
account the order of the draw. They are called the ordered estimates. The order
of the value obtained at previous draw will affect the unbiasedness of
population mean.
- Des Raj ordered estimator
UNORDERED ESTIMATOR:
In ordered estimator, the order in which the units are drawn is considered. Corresponding to any ordered estimator, there exist an unordered estimator which does not depend on the order in which the units are drawn and has smaller variance than the ordered estimator.
- Murthy’s unordered estimator
- Horvitz Thompson (HT) Estimator
- Yates and Grundy (YT) Estimate
HORVITZ THOMPSON (HT) ESTIMATE:
The unordered estimates have limited applicability as they
lack simplicity and the expressions for the estimators and their variance
becomes unmanageable when sample size is even moderately large. The HT estimate
is simpler than other estimators. Let N be the population size and , ( 1,
2,..., ) i yi N = be
the value of characteristic understudy and a sample of size n is drawn by WOR
using arbitrary probability of selection at each draw.
Thus prior to each succeeding draw, there is defined a new
probability distribution for the units available at that draw. The probability
distribution at each draw may or may not depend upon the initial probability at
the first draw.
DRAWBACK:
It does not reduces to zero when all yi pi are same, i.e., when . yi µ pi . Consequently, this may assume negative values for some samples.
YATES AND GRUNDY FORM OF VARIANCE
The Sen-Yates-Grundy estimate for the
variance of a certain estimate of the mean of a finite population is shown to
be admissible in the class of unbiased estimates if the sampling design is of
fixed size two. The admissibility within the unbiased class of three estimates
proposed by Murthy for ordered samples of size two, two for the population mean
and the third for the variance of one of them, is shown to follow by a similar
argument.
APPLICATION:
PPSWOR SAMPLING
(OBJECTIVE):
To give estimates for the average number of
bearing lime tress using exact variance estimates- Horvitz-Thompson (HT) and
Yates-Grundy(yt) and suggest which estimator is more efficient.
DATA DESCRIPTION:
The data is a sample of size 22 from a
survey conduted in different villages growing lime trees in a tehsil of
Bangalore district.
R-CODES:
CONCLUSION:
From the above R code we came to know that the standard error of the estimate by HT estimator is 211.689 and the standard error of the estimate by YT estimator is 27.239
As the standard error of the estimate by Yates and Grundy estimator is less than that of the estimate by Horvitz-Thompson (HT) estimator, Yates and Grundy gives a more efficient estimate.
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