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

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 $T(P,\mathbf{X}_{n})$

Minimax Estimation of Kernel Mean Embeddings

In this paper, we study the minimax estimation of the Bochner integral $$\mu_k(P):=\int_{\mathcal{X}} k(\cdot,x)\,dP(x),$$ also called as the kernel mean embedding, based on random samples drawn i.i.d.~from $P$, where $k:\mathcal{X}\times\mathcal{X}\rightarrow\mathbb{R}$ is a positive definite kernel. Various estimators (including the empirical estimator), $\hat{\theta}_n$ of $\mu_k(P)$ are studied in the literature wherein all of them satisfy $\bigl\| \hat{\theta}_n-\mu_k(P)\bigr\|_{\mathcal{H}_k}=O_P(n^{-1/2})$ with $\mathcal{H}_k$ being the reproducing kernel Hilbert space induced by $k$. The main contribution of the paper is in showing that the above mentioned rate of $n^{-1/2}$ is minimax in $\|\cdot\|_{\mathcal{H}_k}$ and $\|\cdot\|_{L^2(\mathbb{R}^d)}$-norms over the class of discrete measures and the class of measures that has an infinitely differentiable density, with $k$ being a continuous translation-invariant kernel on $\mathbb{R}^d$. The interesting aspect of this result is that the minimax rate is independent of the smoothness of the kernel and the density of $P$ (if it exists). This result has practical consequences in statistical applications as the mean embedding has been widely employed in non-parametric hypothesis testing, density estimation, causal inference and feature selection, through its relation to energy distance (and distance covariance)