Survey Sampling Theory and Methods Second Edition | Date: 28 April 2011, 06:29
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It is gratifying that our Publishers engaged us to prepare this
second edition. Since our first edition appeared in 1992,
Survey Sampling acquired a remarkable growth to which we,
too, have made a modest contribution. So, some addition seems
due. Meanwhile, we have received feedback from our readers
that prompts us to incorporate some modifications.
Several significant books of relevance have emerged after
our write-up for the first edition went to press that we may
now draw upon, by the following authors or editors: SA? RNDAL,
SWENSSON and WRETMAN (1992), BOLFARINE and ZACKS
(1992), S. K. THOMPSON (1992), GHOSH and MEEDEN (1986),
THOMPSON and SEBER (1996), M. E. THOMPSON, (1997)
GODAMBE (1991), COX (1991) and VALLIANT, DORFMAN and
ROYALL (2000), among others.
Numerous path-breaking research articles have also
appeared in journals keeping pace with this phenomenal
progress. So, we are blessed with an opportunity to enlighten
ourselves with plenty of new ideas. Yet we curb our impulse to
cover the salient aspects of even a sizeable section of this current
literature. This is because we are not inclined to reshape
the essential structure of our original volume and we are aware
of the limitations that prevent us from such a venture.
As in our earlier presentation, herein we also avoid being
dogmatic—more precisely, we eschew taking sides. Survey
Sampling is at the periphery of mainstream statistics. The
speciality here is that we have a tangible collection of objects
with certain features, and there is an intention to pry into
them by getting hold of some of these objects and attempting
an inference about those left untouched. This inference
is traditionally based on a theory of probability that is used
to exploit a possible link of the observed with the unobserved.
This probability is not conceived as in statistics, covering other
fields, to characterize the interrelation of the individual values
of the variables of our interest. But this is created by a
survey sampling investigator through arbitrary specification
of an artifice to select the samples from the populations of
objects with preassigned probabilities. This is motivated by
a desire to draw a representative sample, which is a concept
yet to be precisely defined. Purposive selection (earlier purported
to achieve representativeness) is discarded in favor of
this sampling design-based approach, which is theoretically
admitted as a means of yielding a legitimate inference about
an aggregate from a sampled segment and also valued for its
objectivity, being free of personal bias of a sampler. NEYMAN’s
(1934) pioneering masterpiece, followed by survey sampling
texts byYATES (1953),HANSEN,HURWITZ andMADOW (1953),
DEMING (1954) and SUKHATME (1954), backed up by exquisitely
executed survey findings by MAHALANOBIS (1946) in
India as well as by others in England and the U.S., ensured
an unstinted support of probability sampling for about
35 years.
But ROYALL (1970) and BREWER (1963) installed a rival
theory dislodging the role of the selection probability as an
inferential tool in survey sampling. This theory takes off postulating
a probability model characterizing the possible links
among the observed and the unobserved variate values associated
with the survey population units. The parameter of the
surveyor’s inferential concern is now a random variable rather
than a constant. Hence it can be predicted, not estimated.
The basis of inference here is this probability structure as
modeled.
Fortunately, the virtues of some of the sampling designsupported
techniques like stratification, ratio method of
estimation, etc., continue to be upheld by this model-based
prediction theory as well. But procedures for assessing and
measuring the errors in estimation and prediction and setting
up confidence intervals do not match.
The design-based approach fails to yield a best estimator
for a total free of design-bias. By contrast, a model-specific
best predictor is readily produced if the model is simple, correct,
and plausible. If the model is in doubt one has to strike
a balance over bias versus accuracy. A procedure that works
well even with a wrong model and is thus robust is in demand
with this approach. That requires a sample that is adequately
balanced in terms of sample and population values of one or
more variables related to one of the primary inferential interest.
For the design-based classical approach, currently recognized
performers are the estimators motivated by appropriate
prediction models that are design-biased, but the biases
are negligible when the sample sizes are large. So, a modern
compromise survey approach called model assisted survey
sampling is now popular. Thanks to the pioneering efforts by
SA? RNDAL (1982) and his colleagues the generalized regression
(GREG) estimators of this category are found to be very effective
in practice.
Regression modeling motivated their arrival. But an alternative
calibration approach cultivated since the early
nineties by ZIESCHANG (1990),DEVILLE and SA? RNDAL (1992),
and others renders them purely design-based as well with an
assured robustness or riddance from model-dependence
altogether.
A predictor for a survey population total is a sum of
the sampled values plus the sum of the predictors for the
unsampled ones. A design-based estimator for a population
total, by contrast, is a sum of the sampled values with multiplicative
weights yielded by specific sampling designs. A calibration
approach adjusts these initial sampling weights, the
new weights keeping close to them but satisfying certain
consistency constraints or calibration equations determined
by one or more auxiliary variables with known population
totals.
This approach was not discussed in the first edition but is
now treated at length. Adjustments here need further care to
keep the new weights within certain plausible limits, for which
there is considerable documentation in the literature. Here we
also discuss a concern for outliers—a topic which also recommends
adjustments of sampling weights. While calibration and
restricted calibration estimators remain asymptotically design
unbiased (ADU) and asymptotically design consistent (ADC),
the other adjusted ones do not.
Earlier we discussed the QR predictors, which include
(1) the best predictors, (2) projection estimators, (3) generalized
regression estimators, and (4) the cosmetic predictors
for which (1) and (3) match under certain conditions. Developments
since 1992 modify QR predictors into restricted QR
predictors (RQR) as we also recount.
SA? RNDAL (1996), DEVILLE (1999), BREWER (1999a,
1999b), and BREWER and GREGOIRE (2000) are prescribing a
line of research to justify omission of the cross-product terms
in the quadratic forms, giving the variance and mean square
error (MSE) estimators of linear estimators of population totals,
by suitable approximations. In this context SA? RNDAL
(1996) makes a strong plea for the use of generalized regression
estimators based either on stratified (1) simple random
sampling (SRS) or (2) Bernoulli sampling (BS), which is a
special case of Poisson sampling devoid of cross-product
terms. This encourages us to present an appraisal of
Poisson sampling and its valuable ramifications employing
permanent random numbers (PRN), useful in coordination and
exercise of control in rotational sampling, a topic we omitted
earlier.
Among other novelties of this edition we mention the following.
We give essential complements to our earlier discussion
of the minimax principle. In the first edition, exact results
were presented for completely symmetric situations and approximate
results for large populations and samples. Now, following
STENGER and GABLER (1996) an exact minimax
property of the expansion estimator in connection with the
LAHIRI-MIDZUNO-SEN design is presented for arbitrary sa
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