Size constrained unequal probability sampling with a noninteger sum of inclusion probabilities
A. Grafström, Lionel Qualité, Yves Tillé & Alina Matei
Abstract 
More than 50 methods have been developed to draw unequal probability
samples with fixed sample size. All these methods require the sum of
the inclusion probabilities to be an integer number. There are
cases, however, where the sum of desired inclusion probabilities is
not an integer. Then, classical algorithms for drawing samples
cannot be directly applied. We present two methods to overcome the
problem of sample selection with unequal inclusion probabilities
when their sum is not an integer and the sample size cannot be
fixed. The first one consists in splitting the inclusion
probability vector. The second method is based on extending the
population with a phantom unit. For both methods the sample size is
almost fixed, and equal to the integer part of the sum of the
inclusion probabilities or this integer plus one. 
Keywords 

Citation  Grafström, A., Qualité, L., Tillé, Y., & Matei, A. (2012). Size constrained unequal probability sampling with a noninteger sum of inclusion probabilities. Electronic Journal of Statistics, 6, 14771489. 
Type  Journal article (English) 
Date of appearance  2012 
Journal  Electronic Journal of Statistics 
Volume  6 
Pages  14771489 
URL  http://projecteuclid.org/DPubS?verb=Display&version=1.0&s... 