Risk Bounds in Bivariate Isotonic Regression

• Activity Number: 202
• Type: Invited
• Date/Time: Monday, August 4, 2014 : 2:00 PM to 3:50 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Adityanand Guntuboyina*+
• Companies: University of California, Berkeley
• Keywords: isotonic regression ; adaptive estimation ; variable adaptation ; multidimensional shape constrained estimation ; least squares estimator

Abstract

We consider estimation of an unknown regression function $f$, defined on the two-dimensional unit square, that is non-decreasing in each coordinate. This is the bivariate isotonic regression problem. We work under the fixed design setting where $f$ is observed with gaussian noise at fixed grid points of the unit square. The most natural estimator under this setting is the least squares estimator (LSE). We prove risk bounds for the LSE under the natural squared error loss. An interesting aspect here is that the LSE displays automatic adaptation. Specifically, the LSE converges at the parametric rate (upto logarithmic factors) when f is piecewise constant on a small number of rectangular blocks. Also, when the true function $f (x, y) = g(x)$ for a univariate non-decreasing function $g$, the LSE converges at the univariate rate $n^{-2/3}$. To the best of our knowledge, this is the first time such adaptation and risk bounds are proved for a multidimensional shape constrained estimation problem.