占据数和缺失质量的无限urn格式中的浓度不等式及其应用
Concentration inequalities in the infinite urn scheme for occupancy counts and the missing mass, with applications
摘要：
无限urn方案由正整数上的概率质量函数$(p_j)_{j\geq1}$定义。随机分配由根据该概率分布的$N$独立绘图的样本组成，其中$N$可以是确定性的或泊松分布的。本文关注的是占用计数，即与样本中具有$r$或至少$r$出现的符号的数量，以及缺失质量，即样本中未出现的所有符号的总概率。在不对抽样分布作任何进一步假设的情况下，证明了这些随机量满足Bernstein型集中不等式。这些浓度不等式中的方差因子被证明是紧的，如果抽样分布满足一个经常变化的性质。这个常规的变量属性如下所示。假设概率大于$x$的符号的数目是$\vecν(x)=|\{j:p_j\geq x\}|$。在规则变化的urn方案中，对于$α\in[0,1]$，$\vecν$满足$\lim_{τ\rightarrow0}\vecν(τx)/\vecν(τ)=x^{-α}$，并且随着样本大小趋于无穷大，样本中不同符号的数量的方差趋于无穷大。在其他应用中，这些浓度不等式使我们能够得到紧密的置信区间的良好-图灵估计的丢失的质量。
英文摘要：
An infinite urn scheme is defined by a probability mass function $(p_j)_{j\geq1}$ over positive integers. A random allocation consists of a sample of $N$ independent drawings according to this probability distribution where $N$ may be deterministic or Poisson-distributed. This paper is concerned with occupancy counts, that is with the number of symbols with $r$ or at least $r$ occurrences in the sample, and with the missing mass that is the total probability of all symbols that do not occur in the sample. Without any further assumption on the sampling distribution, these random quantities are shown to satisfy Bernstein-type concentration inequalities. The variance factors in these concentration inequalities are shown to be tight if the sampling distribution satisfies a regular variation property. This regular variation property reads as follows. Let the number of symbols with probability larger than $x$ be $\vecν(x)=|\{j:p_j\geq x\}|$. In a regularly varying urn scheme, $\vecν$ satisfies $\lim_{τ\rightarrow0}\vecν(τx)/\vecν(τ)=x^{-α}$ for $α\in[0,1]$ and the variance of the number of distinct symbols in a sample tends to infinity as the sample size tends to infinity. Among other applications, these concentration inequalities allow us to derive tight confidence intervals for the Good--Turing estimator of the missing mass.
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