In this talk we consider a general question of estimating linear functional of the distribution based on the noisy samples from it. We discover that the (two-point) LeCam lower bound is in fact achievable by optimizing bias-variance tradeoff of an empirical-mean type of estimator. We extend the method to certain symmetric functionals of high-dimensional parametric models.

Next, we apply this general framework to two problems: population recovery and predicting the number of unseen species. In population recovery, the goal is to estimate an unknown high-dimensional distribution (in $L_\infty$-distance) from noisy samples. In the case of \textit{erasure} noise, i.e. when each coordinate is erased with probability $\epsilon$, we discover a curious phase transition in sample complexity at $\epsilon=1/2$. In the second (classical) problem, we observe $n$ iid samples from an unknown distribution on a countable alphabet and the goal is to predict the number of new species that will be observed in the next (unseen) $tn$ samples. Again, we discover a phase transition at $t=1$. In both cases, the complete characterization of sample complexity relies on complex-analytic methods, such as Hadamard's three-lines theorem.

Joint work with Yihong Wu (Yale)