Speaker
Description
Consider the broken random sample problem introduced by DeGroot, Feder, and Gole (1971 Ann. Math. Statist.): In each observation, a random sample of M point pairs ((Xᵢ,Yᵢ))ᵢ₌₁ᴹ is drawn from a joint distribution with density f(x,y). Assume we can only observe (Xᵢ)ᵢ₌₁ᴹ and (Yᵢ)ᵢ₌₁ᴹ separately, i.e.~the pairing information is lost. We are interested in estimating f given N i.i.d.~observations. Naive maximization of the likelihood quickly becomes numerically intractable as M increases, due to the combinatorial number of possible pairings between the points. Instead, we propose a quasi-likelihood and provide convergence results for the corresponding estimator. Moreover, our ongoing research shows that under some mild assumptions, the convergence is uniform in M. This allows estimation of f even when the ``density" of the observed point clouds is very high. Applications of the broken sample setting are colocalization analysis in fluorescent microscopy and dynamic systems where we obtain encouraging results in numerical experiments.
