Optimal estimation of high-order missing masses, and the rare-type match problem

TL;DR

This paper develops optimal nonparametric estimators for high-order missing masses in discrete distributions, analyzes their minimax properties, and applies these results to forensic likelihood ratio estimation.

Contribution

It introduces a new estimator for the r-order missing mass, establishes conditions for its minimax optimality, and connects the estimation to forensic likelihood ratio problems.

Findings

The estimator is consistent under certain tail conditions.

Minimax estimation is impossible over all discrete distributions.

The estimator achieves minimax optimality under stronger tail assumptions.

Abstract

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 a stronger assumption 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 $T(P,\mathbf{X}_{n})$.