Sampling Designs - Sampling Spatial Units for Agricultural Surveys - page 142

implicit . This means that the statistical consequences of the restrictions to the frame
have long been ignored.
Cut-off sampling is, in some sense, in an intermediate position between proba-
bilistic and non-probabilistic sampling schemes. This is a feature that is not
appreciated by survey methodologists. As a result, there are very few papers
concerning its theoretical foundations.
It is worth mentioning at least four practical advantages of cut-off sampling in
relation to the costs of a survey (Benedetti et al. 2010 ):

1. Building and updating a sampling frame for small business units could be too
costly, because the gain in efficiency of the estimators would probably be small.
2. Excluding the units of the population that have little contribution to the aggre-
gates under investigation usually results in a large decrease in the number of
units that have to be surveyed to get a predefined accuracy level.
3. Constraining the frame and, as a consequence, the sample, reduces the problem
of empty strata that mainly affects the smallest units. The non-response rate, the
turnover rate of economic units, and the errors of under- or over-coverage of the
frame become more relevant as the size of the units decreases.
4. Cut-off sampling may be demonstrably preferable in terms of accuracy when the
total survey error is taken into account.

Given that practitioners are in favor of such partitions of the population, and
there are technical reasons that justify their use, we wonder whether it is possible to
consider cut-off sampling as a valid sampling scheme.
Assume that we are interested in estimating the total of a population. Benedetti
et al. ( 2010 ) developed a computationally feasible solution for constructing the
three strata in a multipurpose and multivariate setup. The estimator of the total can
be written as

!

t C þt S X

X

þʴ þʴ

t y ¼

t C þt S þt E ¼ 1 ð Þ ¼ 1 y k þ d k y k ; ð 6 : 26 Þ

k 2 U C k 2 s

where ʴ is given by ʴ ¼ π k are the direct sampling

weights of the design used in U S . The optimal sample size n is given by

t x , U E = ð t x , U C þ t x , U S Þ , and d k ¼ 1/

1

n ¼ N N E ; ð 6 : 27 Þ

S y , U S
ˈ

1

N S þ