Density estimation from batched broken random samples

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.