Hang Deng is a PhD candidate in statistics at Rutgers University, working under the supervision of Cun-Hui Zhang. Currently, his research mainly focus on (i) high dimensional central limit theorem and bootstrap methods for the statistical inference of high dimensional models and (ii) nonparametric methods for statistical learning under qualitative shape restrictions. He received his B.Sc. degree in Mathematics and Applied Mathematics from Fudan University.
Talk: Estimation and Inference in Multiple Isotonic Regression and Other Shape Constrained Problems
A link to this Zoom talk will be sent to the Stats Seminar list serv
Abstract: Shape constraints such as monotonicity, convexity and log-concavity can naturally emerge from real applications or are imposed to replace parametric assumptions to allow more flexibility. Estimation and inference under shape constraints have been shown to be useful in genetics, survival analysis, dose-response modeling, economics, etc. In this talk, I will mainly focus on the multivariate case of isotonic regression where the unknown mean function is coordinate-wise nondecreasing.
We first study minimax and adaptation rates in general isotonic regression. The common least squares estimator (LSE) is known to nearly achieve the minimax rate for the $\ell_2$ risk but it exhibits strictly sub-optimal adaptation behavior to piecewise constant mean functions and variable selection in most settings. In contrast, we develop a class of estimators, called the block estimators, and show that they not only yield better risk rates than the existing ones for the LSE to achieve rate optimality, but also adapt to piecewise constant functions at parametric rate and to variable selection at oracle minimax rate up to a logarithmic factor.
We then study the problem of constructing pointwise confidence intervals using a block estimator. We propose the innovative idea of local normalization and construct the \textit{first tuning-free} confidence interval for this problem. Such confidence interval is shown to have asymptotically exact confidence level and oracle length. The idea can be used to develop similar methods for other shape constrained inference problems including the widely studied log-concave density estimation.