A Compound Poisson Perspective of Ewens–Pitman Sampling Model

The Ewens–Pitman sampling model (EP-SM) is a distribution for random partitions of the set {1,…,n}, with n∈N, which is indexed by real parameters α and θ such that either α∈[0,1) and θ>−α, or α<0 and θ=−mα for some m∈N. For α=0, the EP-SM is reduced to the Ewens sampling model (E-SM), which admits a well-known compound Poisson perspective in terms of the log-series compound Poisson sampling model (LS-CPSM). In this paper, we consider a generalisation of the LS-CPSM, referred to as the negative Binomial compound Poisson sampling model (NB-CPSM), and we show that it leads to an extension of the compound Poisson perspective of the E-SM to the more general EP-SM for either α∈(0,1), or α<0. The interplay between the NB-CPSM and the EP-SM is then applied to the study of the large n asymptotic behaviour of the number of blocks in the corresponding random partitions—leading to a new proof of Pitman’s α diversity. We discuss the proposed results and conjecture that analogous compound Poisson representations may hold for the class of α-stable Poisson–Kingman sampling models—of which the EP-SM is a noteworthy special case.

On four measures of taxonomic richness

The choice of measures used to estimate the richness of species, genera, or higher taxa is a crucial matter in paleobiology and ecology. This paper evaluates four methods called shareholder quorum subsampling, true richness estimated using a Poisson sampling model (TRiPS), squares, and the corrected first-order jackknife (cJ1). Quorum subsampling interpolates to produce a relative richness estimate, while the other three extrapolate to the size of the overall species pool. Here I use routine ecological data to show that squares and cJ1 pass several basic validation tests, but TRiPS does not. First, TRiPS estimates are insensitive to the shape of abundance distributions, being entirely predicted by total counts of species and of individuals regardless of the details. Furthermore, TRiPS tends not to extrapolate at all when sampling is moderate or intense. Second, all three extrapolators yield lower values when they work with small uniform subsamples of large raw inventories. The third test is a split-analyze-and-sum analysis: each inventory is divided between the most common and least common halves of the abundance distribution, the methods are applied to the half-inventories, and the estimates are summed. Squares and cJ1 perform well here, but TRiPS does not extrapolate as long as the full inventories are reasonably well-sampled. It is otherwise not particularly accurate. The extrapolators are largely insensitive to the influence of abundance distribution evenness, as quantified using Pielou's J and a new index called the ratio of means. Quorum subsampling generally performs well, but it stumbles on the split-analyze-and-sum test and is confounded somewhat by evenness.

A metodologia de amostragem do Saeb

Apresenta, sob um enfoque simplificado, a metodologia de amostragem do Saeb, enfatizando particularidades relacionadas com o ciclo de 2001 e buscando esclarecer o leitor em relação a alguns dos pontos considerados mais importantes, tais como população de referência, estágios, esquemas de seleção de unidades amostrais e métodos de análise de dados de amostras complexas. Tópicos como precisão amostral, que envolvem discussões corriqueiras sobre a não-publicação de resultados individuais por escola ou a repetição de escolas que participaram do Saeb 1999, também são discutidos. Citam-se, ainda, casos de institutos de pesquisa sediados em outros países e que utilizam metodologias de amostragem semelhantes a do Saeb. Por fim, são apresentadas sugestões para o Saeb em suas futuras realizações. Palavras-chave: amostras complexas; coordenação de amostras; amostragem seqüencial de Poisson; números aleatórios permanentes. Abstract This paper presents a simplified version of the Saeb sampling methodology, emphasizing particularities of its cycle of 2001, and aiming at making clear to the reader some of its main points, such as target population, stages, sampling schemes and data analyses methods applied to complex samples. Topics like precision of estimates, that involve customary discussions about the avoidance in publishing schools' individual results or the sample overlap with the Saeb 1999 are also discussed. Cases of international research institutes that make use of sampling methodologies similar to that of Saeb are also mentioned. To finish, we present suggestions to Saeb in its future realizations. Keywords: complex samples; sample overlap; sequential Poisson sampling; permanent random numbers.