The broken sample problem revisited: Proof of a conjecture by Bai-Hsing and high-dimensional extensions

Abstract

We revisit the classical broken sample problem: Two samples of i.i.d.\ data points $\mathbf{X}=\{X_1,\dots ,X_n\}$ and $\mathbf{Y}=\{Y_1,\dots ,Y_m\}$ are observed without correspondence with $m\le n$. Under the null hypothesis, $\mathbf{X}$ and $\mathbf{Y}$ are independent. Under the alternative hypothesis, $\mathbf{Y}$ is correlated with a random subsample of $\mathbf{X}$, in the sense that $(X_{\ddot{I}€(i)},Y_i)$'s are drawn independently from some bivariate distribution for some latent injection $\ddot{I}€:[m]\to [n]$. Originally introduced by DeGroot, Feder, and Goel to model matching records in census data, this problem has recently gained renewed interest due to its applications in data de-anonymization, data integration, and target tracking. Despite extensive research over the past decades, determining the precise detection threshold has remained an open problem even for equal sample sizes ($m=n$). Assuming $m$ and $n$ grow proportionally, we show that the sharp threshold is given by a spectral and an $L_2$ condition of the likelihood ratio operator, resolving a conjecture of Bai and Hsing in the positive. These results are extended to high dimensions and settle the sharp detection thresholds for Gaussian and Bernoulli models.

38 pages, 3 figures