We study limit distributions for the tuning-free max-min block estimator originally proposed in Fokiano et al. (2017) in the problem of multiple isotonic regression, under both fixed lattice design and random design settings. There are two interesting features in our local theory. First, the max-min block estimator automatically adapts to the full spectrum of local smoothness levels and the intrinsic dimension of the isotonic regression function at the optimal rate. Second, the optimally adaptive local rates are in general not the same in fixed lattice and random designs. In fact, the local rate in the fixed lattice design case is no slower than that in the random design case, and can be much faster when the local smoothness levels of the isotonic regression function or the sizes of the lattice differ substantially along different dimensions.
The talk is based on joint work with Cun-Hui Zhang.
About the Speaker
Qiyang Han is an assistant professor of Statistics at Rutgers University. He received a Ph.D. in Statistics from University of Washington in 2018 under the supervision of Professor Jon A. Wellner. He is broadly interested in mathematical statistics and high dimensional probability. His current research is concentrated on abstract empirical process theory, and its applications to nonparametric function estimation (with a special focus on shape-restricted problems), Bayes nonparametrics, and high dimensional statistics.