Density estimation from batched broken random samples

TL;DR

This paper introduces a parametric estimation method for density functions from broken random samples where pairing information is lost, demonstrating fast convergence rates with increasing sample batches.

Contribution

It proposes a pseudo-log-likelihood based estimator for density from broken samples and proves its fast convergence rate independent of batch size.

Findings

Estimator achieves fast convergence rate in number of batches

Method works under mild assumptions

Convergence rate is uniform in batch size

Abstract

The broken random sample problem was first introduced by DeGroot, Feder, and Gole (1971, Ann. Math. Statist.): in each observation (batch), a random sample of $M$ i.i.d. point pairs $ ((X_i,Y_i))_{i=1}^M$ is drawn from a joint distribution with density $p(x,y)$, but we can observe only the unordered multisets $(X_i)_{i=1}^M$ and $(Y_i)_{i=1}^M$ separately; that is, the pairing information is lost. For large $M$, inferring $p$ from a single observation has been shown to be essentially impossible. In this paper, we propose a parametric method based on a pseudo-log-likelihood to estimate $p$ from $N$ i.i.d. broken sample batches, and we prove a fast convergence rate in $N$ for our estimator that is uniform in $M$, under mild assumptions.