The problem of species richness estimation using complete monotonicity of the distribution of species abundances is considered. Complete monotonicity is the most natural surrogate for k-monotonicity when k is large. The latter model has been considered in the same estimation problem adopting two different approaches which both necessitate selecting the unknown degree of monotonicity k via some chosen criterion. It is shown that such selection procedures can be avoided by appropriately approximating the true completely monotone distribution by a kn-monotone one such that kn grows logarithmically as a function of the sample size n. Furthermore, the proposed estimator of the true total number of species is proved to be asymptotically normal. An extended simulation study indicates that it is quite competitive when compared to other available estimators, and this remains true even when complete monotonicity is not satisfied. It is further illustrated how the method can be applied in practice by using four real datasets.

中文翻译：

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完全单调分布：物种数量的混合、近似和估计

考虑了使用物种丰度分布的完全单调性来估计物种丰富度的问题。当 k 很大时，完全单调性是 k 单调性的最自然替代品。后一种模型已在采用两种不同方法的同一估计问题中得到考虑，这两种方法都需要通过某些选定的标准来选择未知的单调度 k。结果表明，通过用 kn 单调分布适当地逼近真正的完全单调分布，使得 kn 作为样本大小 n 的函数以对数方式增长，可以避免这种选择过程。此外，提议的真实物种总数估计量被证明是渐近正态的。扩展的模拟研究表明，与其他可用的估计量相比，它具有相当的竞争力，即使不满足完全单调性，这仍然是正确的。通过使用四个真实数据集，进一步说明了该方法如何在实践中应用。