On Some Problems in Nonparametric and Location-Scale Estimation

We study three generalizations of classical nonparametric, and locationscale estimation problems. 

First, we study the classical problem of deriving minimax rates for density estimation over convex density classes. Our work extends known results by demonstrating that the local metric entropy of the density class always captures the exact (up to constants) minimax optimal rates under such settings. Our bounds provide a unifying perspective across both parametric and nonparametric convex density classes, under weaker assumptions on the richness of the density class than previously considered. 

Second, we consider a variation of classical isotonic regression, which we term adversarial sign-corrupted isotonic (ASCI) regression. Here, the adversary can corrupt the sign of the responses having full access to the true response terms. We formalize ASCIFIT, a three-step estimation procedure under this regime, and demonstrate its theoretical guarantees in the form of sharp high probability upper bounds and minimax lower bounds.

Finally, we extend classical univariate uniform location-scale estimation over an interval, to multivariate uniform location-scale estimation over general convex bodies. Unlike the univariate setting, the observations are no longer totally ordered, and previous estimation techniques prove insufficient to account for the more refined geometry of the generating process. Under fixed dimension, our proposed location estimators converge at an n^-1 rate. Our minimax lower bounds justify the optimality of our estimators in terms of the sample complexity. We also provide practical algorithms with provable convergence rates for our estimators, over a wide class of convex bodies.