Consider a random sample $(X_{1},\ldots,X_{n})$ from an unknown discrete distribution $P=\sum_{j\geq1}p_{j}\delta_{s_{j}}$ on a countable alphabet $\mathbb{S}$, and let $(Y_{n,j})_{j\geq1}$ be the empirical frequencies of distinct symbols $s_{j}$'s in the sample. We consider the problem of estimating the $r$-order missing mass, which is a discrete functional of $P$ defined as $$\theta_{r}(P;\mathbf{X}_{n})=\sum_{j\geq1}p^{r}_{j}I(Y_{n,j}=0).$$ This is generalization of the missing mass whose estimation is a classical problem in statistics, being the subject of numerous studies both in theory and methods. First, we introduce a nonparametric estimator of $\theta_{r}(P;\mathbf{X}_{n})$ and a corresponding non-asymptotic confidence interval through concentration properties of $\theta_{r}(P;\mathbf{X}_{n})$. Then, we investigate minimax estimation of $\theta_{r}(P;\mathbf{X}_{n})$, which is the main contribution of our work. We show that minimax estimation is not feasible over the class of all discrete distributions on $\mathbb{S}$, and not even for distributions with regularly varying tails, which only guarantee that our estimator is consistent for $\theta_{r}(P;\mathbf{X}_{n})$. This leads to introduce the stronger assumption of second-order regular variation for the tail behaviour of $P$, which is proved to be sufficient for minimax estimation of $\theta_r(P;\mathbf{X}_{n})$, making the proposed estimator an optimal minimax estimator of $\theta_{r}(P;\mathbf{X}_{n})$. Our interest in the $r$-order missing mass arises from forensic statistics, where the estimation of the $2$-order missing mass appears in connection to the estimation of the likelihood ratio $T(P,\mathbf{X}_{n})=\theta_{1}(P;\mathbf{X}_{n})/\theta_{2}(P;\mathbf{X}_{n})$, known as the "fundamental problem of forensic mathematics". We present theoretical guarantees to nonparametric estimation of $T(P,\mathbf{X}_{n})$.

翻译：考虑来自可数字母表 $\mathbb{S}$ 上未知离散分布 $P=\sum_{j\geq1}p_{j}\delta_{s_{j}}$ 的随机样本 $(X_{1},\ldots,X_{n})$，并令 $(Y_{n,j})_{j\geq1}$ 为样本中不同符号 $s_{j}$ 的经验频率。我们研究 $r$ 阶缺失质量的估计问题，该量定义为 $$ heta_{r}(P;\mathbf{X}_{n})=\sum_{j\geq1}p^{r}_{j}I(Y_{n,j}=0)，$$ 是 $P$ 的离散泛函。这是缺失质量的推广形式，其估计是统计学中的经典问题，在理论和方法上均有广泛研究。首先，我们基于 $ heta_{r}(P;\mathbf{X}_{n})$ 的集中性质，引入该量的非参数估计量及相应的非渐近置信区间。随后，我们研究 $ heta_{r}(P;\mathbf{X}_{n})$ 的极小极大估计，这是本文的主要贡献。我们证明，对于 $\mathbb{S}$ 上所有离散分布构成的类，甚至对于仅能保证估计量对 $ heta_{r}(P;\mathbf{X}_{n})$ 具有一致性的正则变化尾部分布，极小极大估计均不可行。这促使我们引入更强的假设——$P$ 的尾部行为具有二阶正则变化性，并证明该条件足以实现 $ heta_{r}(P;\mathbf{X}_{n})$ 的极小极大估计，使所提出的估计量成为 $ heta_{r}(P;\mathbf{X}_{n})$ 的最优极小极大估计。我们对 $r$ 阶缺失质量的兴趣源于法医统计学，其中 $2$ 阶缺失质量的估计与似然比 $T(P,\mathbf{X}_{n})= heta_{1}(P;\mathbf{X}_{n})/ heta_{2}(P;\mathbf{X}_{n})$（被称为“法医数学基本问题”）的估计相关联。我们为 $T(P,\mathbf{X}_{n})$ 的非参数估计提供了理论保障。