31: Learning Smooth Populations of Parameters with Trial Heterogeneity

Music City Center 

We revisit the classical problem of estimating the mixing distribution of Binomial mixtures under trial heterogeneity and smoothness. This problem has been studied extensively when the trial parameter is homogeneous, but not under the more realistic scenario of heterogeneous trials, and only within a low smoothness regime, where the resulting rates are suboptimal. Under the assumption that the density is s-smooth, we derive faster error rates for nonparametric density estimators under trial heterogeneity. Importantly, even when reduced to the homogeneous case, our result improves upon the state of the art. We further discuss data-driven tuning parameter selection via cross-validation and a measure of a difference between two densities. Our work is motivated by an application in criminal justice: assessing the effectiveness of indigent representation in Pennsylvania. We find that the estimated conviction rates for appointed counsel (court-appointed private attorneys) are generally higher than those for public defenders, potentially due to a confounding factor: appointed counsel are more likely to take on severe cases.

Binomial mixtures

mixing distribution

nonparametric density estimation

criminal justice 

Section on Nonparametric Statistics