This book is based upon courses on sampling algorithms. After having used
scattered notes for several years, I have decided to completely rewrite the
material in a consistent way. The books of Brewer and Hanif (1983) and
H´ajek (1981) have been my works of reference. Brewer and Hanif (1983) have
drawn up an exhaustive list of sampling methods with unequal probabilities,
which was probably a very tedious work. The posthumous book of H´ajek
(1981) contains an attempt at writing a general theory for conditional Poisson
sampling. Since the publication of these books, things have been improving.
New techniques of sampling have been proposed, to such an extent that it
is difficult to have a general idea of the interest of each of them. I do not
claim to give an exhaustive list of these new methods. Above all, I would
like to propose a general framework in which it will be easier to compare
existing methods. Furthermore, forty-six algorithms are precisely described,
which allows the reader to easily implement the described methods.
This book is an opportunity to present a synthesis of my research and
to develop my convictions on the question of sampling. At present, with the
splitting method, it is possible to construct an infinite amount of new sampling
methods with unequal probabilities. I am, however, convinced that conditional
Poisson sampling is probably the best solution to the problem of sampling with
unequal probabilities, although one can object that other procedures provide
very similar results.
Another conviction is that the joint inclusion probabilities are not used for
anything. I also advocate for the use of the cube method that allows selecting
balanced samples. I would also like to apologize for all the techniques that
are not cited in this book. For example, I do not mention all the methods
called “order sampling” because the methods for coordinating samples are
not examined in this book. They could be the topic of another publication.
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