Bayesian estimation procedures for finite populations, single stage designs, and normal populations
Charles D. Palit, Irwin Guttman · Communications in Statistics · 1973
The Bayesian approach to finite population estimation presented in this paper, requires the use of the super-population concept of Fisher (1956), Cochran (1939, 1946), and others. In this framework, the finite population under consideration is assumed to be a random sample from some hypothetical infinite population. Here, we assume that this hypothetical infinite population has a probability distribution function of known form. This latter assumption has been used by Ericson (1969) in a Bayesian framework, and by Kalbfleisch and Sprott (1956) in a fiducial framework. In this paper, we follow the approach of Fisher (1956) and Kalbfleisch and Sprott (1968) and use a two step procedure to obtain finite population estimates. In general terms, this two step procedure is as follows. First, we use the sample drawn from the finite population to make an inference concerning the parameters of the distribution of the superpopulation. For a Bayesian, this means that we find the posterior distribution of the parameters. The posterior for the parameters is then used to obtain the predictive distribution (see for example Guttman (1967), for the finite population quantity we wish to estimate. Using chis Bayesian analysis. then, we adopt Kalbfleisch and Sprott's (1968) approach for estimating the finite population mean from random eamples and stratified random samples drawn without replacement, obtaining results similar to these of Ericson (1969). We extend the approach to cover the design of two phase random and ;two phase stratified random samples. In this paper, we use a normally distributed super-population, but other super-population distributions could be postulated and used. The reader will note the great similarity between the finite population results obtained here and the finite population results produced by the more conventional distribution free freuuency approach to finite population estimation. (See for-example, chapter 2 and 5 of Cochran (1963), and page 90-141 of Hansen, Hurwitz and Madow (1953)).