Bayesian statistical inference for sampling a finite population
\textit{W. A. Ericson} [J. R. Stat. Soc., Ser. B 31, 195-233 (1969; Zbl 0186.519)] and \textit{B. Hoadley} [J. Am. Stat. Assoc. 64, 216-229 (1969; Zbl 0177.223)] showed that if the prior distribution of the number of population variables belonging to the j th category is Dirichlet- multinomial, the posterior distribution of unobserved population variables (given the observed ones) belonging to each category is also of the same type. \textit{A. Scott} [Ann. Math. Stat. 42, 1113-1117 (1971; Zbl 0228.62007)] proved that the centered and rescaled posterior distribution converges to that of a normal random vector for arbitrary priors. In the present paper these results are extended to the case of arbitrary population variables. The author defines the Dirichlet-multinomial process, extends the Ericson-Hoadley theorem and shows that the Dirichlet process is the limit of the Dirichlet-multinomial process. Next, it is proved that the large-sample posterior distribution of the population empirical distribution, centered and rescaled, is a Brownian bridge with a change of time scale. As corollaries, the large-sample Bayes confidence band for the population empirical distribution function and the large- sample Bayes confidence interval for the population mean are obtained.
centered and rescaled posterior distribution
scientific article; zbMATH DE number 3984324