Sampling formulae arising from random Dirichlet populations

Thierry Huillet (1)

1 LPTM - UMR 8089 - Laboratoire de Physique Théorique et Modélisation

Consider the random Dirichlet partition of the interval into $n$ fragments at temperature $\theta >0.$ Some statistical features of this random discrete distribution are recalled, together with explicit results on the law of its size-biased permutation. Using these, pre-asymptotic versions of the Ewens and Donnelly-Tavar\'{e}-Griffiths sampling formulae from finite Dirichlet partitions are computed exactly. From these, new proofs of the usual sampling formulae from random proportions with GEM$\left( \gamma \right) $ distribution are supplied, when considering the Kingman limit $n\uparrow \infty $, $\theta \downarrow 0$ while $n\theta =\gamma >0$ .

Mots clés: Ewens and Donnelly-Tavar\'{e}-Griffiths sampling formulae; Random discrete distribution; Dirichlet partition; size-biased permutation; GEM; Ewens and Donnelly-Tavar\'{e}-Griffiths sampling formulae.

Domaines: Systèmes désordonnés et réseaux de neurones [cond-mat.dis-nn]

Thierry Huillet. Sampling formulae arising from random Dirichlet populations. Communications in Statistics - Theory and Methods, 2005, 34 (5), pp.1019-1040. ⟨hal-00093134⟩