吉布斯-泊松丰度模型抽样产生的占有率分布
Occupancy distributions arising in sampling from Gibbs-Poisson abundance models
摘要：
从仅显示$p\leq k$不同采样物种的$k-$样本中估计未见过物种的数量$n$已受到长期关注。它需要一个物种丰富度模型和一个取样模型。我们从一个离散的iid随机物种丰度模型开始，每个模型都服从吉布斯-泊松分布。从$n-$物种丰度向量中提取的$k-$样品是在将其调节为$k$ %时获得的样品。我们讨论的抽样公式（物种占有分布，频率的频率）在这方面。然后，我们从样本的大小$k$和观测值$P_{n,k}$，不同的采样物种的数量，$n$问题的估计的一些方面。它表明，它总是有意义的研究这些占用问题，从吉布斯-泊松丰度模型的背景下，具有无穷多个物种的人口。从这个扩展，参数$γ$自然出现，这是物种丰富度或多样性的度量。本文重新推导了无穷多个种群的抽样公式，以及不同抽样种群的个数$P_{k}$的分布。本文从样本容量$k$和观测值$P_{k}$两个方面研究了$γ$问题的估计。然后，我们展示了一个大的特殊类的吉布斯-泊松分布具有的属性，从离散丰度模型采样可以等效地被视为一个随机分区的统一，现在在连续的采样问题。当$n$是有限的时，该划分可以建立在通过其部分和对$% n$无限可除iid正随机变量进行归一化的基础上。它表明，在连续的采样过程中，一般应偏置的总长度出现在后者的归一化。本文还提供了一个从从属子的秩归一化跳跃中进行大小偏差抽样的构造，该问题是否存在无穷多个物种。我们用许多例子来说明我们的观点，其中一些是新的。
英文摘要：
Estimating the number $n$ of unseen species from a $k-$sample displaying only $p\leq k$ distinct sampled species has received attention for long. It requires a model of species abundance together with a sampling model. We start with a discrete model of iid stochastic species abundances, each with Gibbs-Poisson distribution. A $k-$sample drawn from the $n-$species abundances vector is the one obtained while conditioning it on summing to $k$% . We discuss the sampling formulae (species occupancy distributions, frequency of frequencies) in this context. We then develop some aspects of the estimation of $n$ problem from the size $k$ of the sample and the observed value of $P_{n,k}$, the number of distinct sampled species. It is shown that it always makes sense to study these occupancy problems from a Gibbs-Poisson abundance model in the context of a population with infinitely many species. From this extension, a parameter $γ$ naturally appears, which is a measure of richness or diversity of species. We rederive the sampling formulae for a population with infinitely many species, together with the distribution of the number $P_{k}$ of distinct sampled species. We investigate the estimation of $γ$ problem from the sample size $k$ and the observed value of $P_{k}$. We then exhibit a large special class of Gibbs-Poisson distributions having the property that sampling from a discrete abundance model may equivalently be viewed as a sampling problem from a random partition of unity, now in the continuum. When $n$ is finite, this partition may be built upon normalizing $% n$ infinitely divisible iid positive random variables by its partial sum. It is shown that the sampling process in the continuum should generically be biased on the total length appearing in the latter normalization. A construction with size-biased sampling from the ranked normalized jumps of a subordinator is also supplied, would the problem under study present infinitely many species. We illustrate our point of view with many examples, some of which being new ones.
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