GEM随机离散分布抽样中的极值和间隙
Extremes and gaps in sampling from a GEM random discrete distribution
摘要：
我们表明，在来自GEM $(0,θ)$随机离散分布的$n$大小的样本中，样本的顺序统计量$X_{1:n} \le \cdots \le X_{n:n}$之间的间隙$G_{i:n}:= X_{n-i+1:n} - X_{n-i:n}$（按照惯例$G_{n:n} := X_{1:n} - 1$）的分布就像独立几何$(i/(i+θ))$变量$G_i$无限序列的第一个$n$项一样。这将最小$X_{1:n}$的已知结果扩展到样本范围内的其他间隙，并意味着最大$X_{n:n}$具有$1 + \sum_{i=1}^n G_i$的分布，因此已知的结果是$X_{n:n}$像$θ\log(n)$一样增长为$n\to\infty$，具有渐近正态分布。其他结果包括大多数已知的公式的确切分布的GEM $(0,θ)$抽样统计，包括Ewens和唐纳利-Tavaré抽样公式。对于双参数GEM $(α,θ)$分布，我们证明了最大值的增长像$n^{α/(1-α)}$的随机倍数，并找到了乘数的极限分布。
英文摘要：
We show that in a sample of size $n$ from a GEM$(0,θ)$ random discrete distribution, the gaps $G_{i:n}:= X_{n-i+1:n} - X_{n-i:n}$ between order statistics $X_{1:n} \le \cdots \le X_{n:n}$ of the sample, with the convention $G_{n:n} := X_{1:n} - 1$, are distributed like the first $n$ terms of an infinite sequence of independent geometric$(i/(i+θ))$ variables $G_i$. This extends a known result for the minimum $X_{1:n}$ to other gaps in the range of the sample, and implies that the maximum $X_{n:n}$ has the distribution of $1 + \sum_{i=1}^n G_i$, hence the known result that $X_{n:n}$ grows like $θ\log(n)$ as $n\to\infty$, with an asymptotically normal distribution. Other consequences include most known formulas for the exact distributions of GEM$(0,θ)$ sampling statistics, including the Ewens and Donnelly--Tavaré sampling formulas. For the two-parameter GEM$(α,θ)$ distribution we show that the maximal value grows like a random multiple of $n^{α/(1-α)}$ and find the limit distribution of the multiplier.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
