  Protection against disclosure is a legal and ethical obligation for agencies
releasing microdata files for public use. Consider a microdata sample of size
$n$ from a finite population of size $\bar{n}=n+\lambda n$, with $\lambda>0$,
such that each record contains two disjoint types of information: identifying
categorical information and sensitive information. Any decision about releasing
data is supported by the estimation of measures of disclosure risk, which are
functionals of the number of sample records with a unique combination of values
of identifying variables. The most common measure is arguably the number
$\tau_{1}$ of sample unique records that are population uniques. In this paper,
we first study nonparametric estimation of $\tau_{1}$ under the Poisson
abundance model for sample records. We introduce a class of linear estimators
of $\tau_{1}$ that are simple, computationally efficient and scalable to
massive datasets, and we give uniform theoretical guarantees for them. In
particular, we show that they provably estimate $\tau_{1}$ all of the way up to
the sampling fraction $(\lambda+1)^{-1}\propto (\log n)^{-1}$, with vanishing
normalized mean-square error (NMSE) for large $n$. We then establish a lower
bound for the minimax NMSE for the estimation of $\tau_{1}$, which allows us to
show that: i) $(\lambda+1)^{-1}\propto (\log n)^{-1}$ is the smallest possible
sampling fraction; ii) estimators' NMSE is near optimal, in the sense of
matching the minimax lower bound, for large $n$. This is the main result of our
paper, and it provides a precise answer to an open question about the
feasibility of nonparametric estimation of $\tau_{1}$ under the Poisson
abundance model and for a sampling fraction $(\lambda+1)^{-1}<1/2$.
