Fisher’s measure of variability in repeated samples

IF 1.5 2区 数学 Q2 STATISTICS & PROBABILITY
Bernoulli Pub Date : 2023-05-01 DOI:10.3150/22-bej1494
Poly H. da Silva, Arash Jamshidpey, P. McCullagh, S. Tavaré
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引用次数: 3

Abstract

Fisher (1943) claimed that the expected value of the sample variance of the number of species found in large samples, each of n specimens taken from the same population, is asymptotically θ log2. This is at odds with the value θ log n obtained directly from the Ewens Sampling Formula (ESF), where θ specifies the rate at which new species are found. To resolve this apparent contradiction, we assume the species frequency spectrum in the population is determined by the ESF and that the samples are disjoint subsets drawn sequentially from this single population. We find an explicit formula for the required expected value for p samples of arbitrary size; in the limit of large equally-sized samples, it indeed has the value θ log2. We obtain limit theorems for the sample variance of p samples of size n under various limiting regimes as p , n or both tend to ∞ . We discuss further the behavior of the number of species present in all samples, and revisit Fisher’s log-series distribution as the limiting distribution of the number of specimens observed in typical species in a future, large sample.
重复样本变异性的Fisher测度
Fisher(1943)声称,在大样本中发现的物种数量的样本方差的期望值是渐近的θ log2,每n个样本取自同一种群。这与直接从埃文斯抽样公式(ESF)得到的θ log n值不一致,在ESF中,θ表示发现新物种的速率。为了解决这个明显的矛盾,我们假设种群中的物种频谱是由ESF决定的,并且样本是从这个单一种群中顺序抽取的不相交的子集。我们找到了任意大小的p个样本所需期望值的显式公式;在大小相等的大样本的极限下,它的值确实是θ log2。当p、n或两者都趋于∞时,我们得到了大小为n的p个样本在各种极限情况下的样本方差的极限定理。我们进一步讨论了所有样本中存在的物种数量的行为,并重新审视Fisher的对数序列分布,作为未来大样本中典型物种中观察到的标本数量的极限分布。
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来源期刊
Bernoulli
Bernoulli 数学-统计学与概率论
CiteScore
3.40
自引率
0.00%
发文量
116
审稿时长
6-12 weeks
期刊介绍: BERNOULLI is the journal of the Bernoulli Society for Mathematical Statistics and Probability, issued four times per year. The journal provides a comprehensive account of important developments in the fields of statistics and probability, offering an international forum for both theoretical and applied work. BERNOULLI will publish: Papers containing original and significant research contributions: with background, mathematical derivation and discussion of the results in suitable detail and, where appropriate, with discussion of interesting applications in relation to the methodology proposed. Papers of the following two types will also be considered for publication, provided they are judged to enhance the dissemination of research: Review papers which provide an integrated critical survey of some area of probability and statistics and discuss important recent developments. Scholarly written papers on some historical significant aspect of statistics and probability.
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