A Bayesian bootstrap for a finite population.

*(English)*Zbl 0691.62005The author introduces a Bayesian bootstrap for a finite population (FPBB, i.e., finite population Bayesian bootstrap) and defines it in terms of the urn of observations and compares it with FPB (finite population bootstrap). In operation, the FPBB is simply a Pólya sampling from the data urn, whereas the FPB is a simple random sampling from a mixture of two preliminary FPB populations. The FPBB is simpler than the FPB in the sense that no randomized population is involved.

On the other hand, the Pólya sample size m \((=N-n)\) could be much larger than the FPB sample size n. The difference then is a randomized FPB population against a perhaps larger Pólya sample size. The paramount distribution in the FPB case is the multivariate hypergeometric distribution, but the author shows that the paramount distribution in the FPBB case is the Dirichlet-multinomial distribution. Moreover, given a sample of size n from the finite population, the FPBB distribution of the standardized unknown empirical distribution of the population converges weakly to a Brownian bridge as the sample size tends to \(\infty\), as long as the sample empirical distribution converges to a distribution function. And, the FPBB and FPB are first-order asymptotically equivalent.

On the other hand, the Pólya sample size m \((=N-n)\) could be much larger than the FPB sample size n. The difference then is a randomized FPB population against a perhaps larger Pólya sample size. The paramount distribution in the FPB case is the multivariate hypergeometric distribution, but the author shows that the paramount distribution in the FPBB case is the Dirichlet-multinomial distribution. Moreover, given a sample of size n from the finite population, the FPBB distribution of the standardized unknown empirical distribution of the population converges weakly to a Brownian bridge as the sample size tends to \(\infty\), as long as the sample empirical distribution converges to a distribution function. And, the FPBB and FPB are first-order asymptotically equivalent.

Reviewer: H.-J.Chang