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Looking-backward probabilities for Gibbs-type exchangeable random partitions. (English) Zbl 1329.60091

The authors are focused on the class of Gibbs-type exchangeable random partitions. These are random partitions which arise by sampling from a random probability measure of Gibbs-type, denoted \(\widetilde P_G\) (see [J. Pitman, “Poisson-Kingman partitions”, in: D. R. Goldstein (ed.), Statistics and science: a Festschrift for Terry Speed. Beachwood, OH: Institute of Mathematical Statistics. 1–34 (2003)]).
A recent trend is the study of their conditional properties, that is, the study of the estimation of the distribution of certain statistics of an additional sample \((X_{n+1},\dots, X_{n+m})\), to the initially given sample \((X_1,\dots, X_n)\) from \(\widetilde P_G\). The authors point out the existence of a plenty of open problems in the conditions analysis of Gibbs-type exchangeable random partitions.
The present study advances some new results, as for instance on the looking-backward species sampling problems in the general case of Gibbs-type exchangeable random partitions and in the special case of the celebrated Ewens-Pitman sample model.
The paper is well-motivated and didactically written.

MSC:

60G09 Exchangeability for stochastic processes
60C05 Combinatorial probability
62G05 Nonparametric estimation
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References:

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