Abstract
Statistical methods to produce inferences based on samples from finite populations have been available for at least 70 years. Topics such as Survey Sampling and Sampling Theory have become part of the mainstream of the statistical methodology. A wide variety of sampling schemes as well as estimators are now part of the statistical folklore. On the other hand, while the Bayesian approach is now a well-established paradigm with implications in almost every field of the statistical arena, there does not seem to exist a conventional procedure—able to deal with both continuous and discrete variables—that can be used as a kind of default for Bayesian survey sampling, even in the simple random sampling case. In this paper, the Bayesian analysis of samples from finite populations is discussed, its relationship with the notion of superpopulation is reviewed, and a nonparametric approach is proposed. Our proposal can produce inferences for population quantiles and similar quantities of interest in the same way as for population means and totals. Moreover, it can provide results relatively quickly, which may prove crucial in certain contexts such as the analysis of quick counts in electoral settings.
Highlights
Survey sampling is one of the most popular areas of Applied Statistics
There were other candidates in that election, but for the sake of simplicity, here we focus on the estimation of the proportion of votes obtained by Diego Sinhue Rodríguez Vallejo (DSRV)
The distribution is unimodal and fairly symmetric, and the 0.95 probability interval is given by [0.4777, 0.5186]. This interval captures rather well the true voting proportion, 0.4998. If these results were used to announce the outcome of the election the very night of the election day, the Quick Count would report that, with probability 0.95, the percentage of votes in favor of DSRV lies between 47.77% and 51.86% or, equivalently, that the percentage of votes in favor of DSRV lies in the interval 49.815% ± 2.045%
Summary
Survey sampling is one of the most popular areas of Applied Statistics. Neyman (1934) [1] established the methodological foundations for statistical inference based on random samples obtained from finite populations. Xin } can be regarded as a set of i.i.d. observations from the superpopulation This framework provides a setting where inferences on the population total T = ∑i∈S xi + ∑i∈/S Xi can be obtained by noting that, upon observing the sample, Tobs = ∑i∈S xi is a known constant, whereas Tnobs = ∑i∈/S Xi can be described through its corresponding posterior predictive distribution. Ericson derived the results for the case of a continuous variable, assuming a normal distribution for the superpopulation He proved that the asymptotic classical results can be obtained as the prior becomes noninformative. Our proposal can deal with both continuous and discrete variables, and can produce inferences for population quantiles and similar quantities of interest in the same way as for population means and totals It can provide results relatively quickly, which may prove crucial in certain contexts such as the analysis of quick counts.
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