Unordered and ordered sample from Dirichlet distribution. (English) Zbl 1095.62013
Summary: Consider the random Dirichlet partition of an interval into \(n\) fragments with parameter \(\theta> 0\). Explicit results on the statistical structure of its size-biased permutation are recalled, leading to (unordered) Ewens and (ordered) Donnelly-Tavaré-Griffiths sampling formulae from finite Dirichlet partitions. We use these preliminary statistical results on frequencies distributions to address the following sampling problem: what are the intervals between new sampled categories when sampling is from Dirichlet populations? The results obtained are in accordance with the ones found in sampling theory from random proportions with GEM\((\gamma)\) distributions. These can be obtained from a Dirichlet model when considering the Kingman limit \(n\uparrow\infty\), \(\theta\downarrow 0\) while \(n\theta= \gamma> 0\).
MSC:
| 62D05 | Sampling theory, sample surveys |
| 62E15 | Exact distribution theory in statistics |
| 60C05 | Combinatorial probability |
Keywords:
random discrete distribution; Dirichlet partition; sizebiased permutation; GEM; Ewens and Donnelly-Tavaré-Griffiths sampling formulae; intervals between new sampled speciesReferences:
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