Topic

# Bernoulli sampling

About: Bernoulli sampling is a(n) research topic. Over the lifetime, 354 publication(s) have been published within this topic receiving 10927 citation(s).

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Abstract: This paper presents a general technique for the treatment of samples drawn without replacement from finite universes when unequal selection probabilities are used. Two sampling schemes are discussed in connection with the problem of determining optimum selection probabilities according to the information available in a supplementary variable. Admittedly, these two schemes have limited application. They should prove useful, however, for the first stage of sampling with multi-stage designs, since both permit unbiased estimation of the sampling variance without resorting to additional assumptions. * Journal Paper No. J2139 of the Iowa Agricultural Experiment Station, Ames, Iowa, Project 1005. Presented to the Institute of Mathematical Statistics, March 17, 1951.

3,642 citations

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Abstract: How many kinds are there? Suppose that a population is partitioned into C classes. In many situations interest focuses not on estimation of the relative sizes of the classes, but on estimation of C itself. For example, biologists and ecologists may be interested in estimating the number of species in a population of plants or animals, numismatists may be concemed with estimating the number of dies used to produce an ancient coin issue, and linguists may be interested in estimating the size of an author's vocabulary. In this article we review the problem of statistical estimation of C. Many approaches have been proposed, some purely data-analytic and others based in sampling theory. In the latter case numerous variations have been considered. The population may be finite or infinite. If finite, samples may be taken with replacement (multinomial sampling) or without replacement (hypergeometric sampling), or by Bernoulli sampling; if infinite, sampling may be multinomial or Bernoulli, or the sample may be th...

712 citations

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Abstract: In [3] the author established necessary and sufficient conditions for asymptotic normality of estimates based on simple random sampling without replacement from a finite population, and thus solved a comparatively old problem initiated by W. G. Madow [8]. The solution was obtained by approximating simple random sampling by so called Poisson sampling, which may be decomposed into independent subexperiments, each associated with a single unit in the population. In the present paper the same method is used for deriving asymptotic normality conditions for a special kind of sampling with varying probabilities called here rejective sampling. Rejective sampling may be realized by $n$ independent draws of one unit with fixed probabilities, generally varying from unit to unit, given the condition that samples in which all units are not distinct are rejected. If the drawing probabilities are constant, rejective sampling coincides, with simple random sampling without replacement, and so the present paper is a generalization of [3]. Basic facts about rejective sampling are exposed in Section 2. To obtain more refined results, Poisson sampling is introduced and analyzed (Section 3) and then related to rejective sampling (Section 4). Next three sections deal with probabilities of inclusion, variance formulas and asymptotic normality of estimators for rejective sampling. In Section 8 asymptotic formulas are tested numerically and applications to sample surveys are indicated. The paper is concluded by short-cuts in practical performance of rejective sampling. The readers interested in applications only may concentrate upon Sections 1, 8 and 9. Those interested in the theory of mean values and variances only, may omit Lemma 4.3 and Section 7.

301 citations