Mathematics > Statistics Theory
[Submitted on 21 Dec 2018 (v1), last revised 10 Jan 2020 (this version, v2)]
Title:Isotonic Regression in Multi-Dimensional Spaces and Graphs
View PDFAbstract:In this paper we study minimax and adaptation rates in general isotonic regression. For uniform deterministic and random designs in $[0,1]^d$ with $d\ge 2$ and $N(0,1)$ noise, the minimax rate for the $\ell_2$ risk is known to be bounded from below by $n^{-1/d}$ when the unknown mean function $f$ is nondecreasing and its range is bounded by a constant, while the least squares estimator (LSE) is known to nearly achieve the minimax rate up to a factor $(\log n)^\gamma$ where $n$ is sample size, $\gamma = 4$ in the lattice design and $\gamma = \max\{9/2, (d^2+d+1)/2 \}$ in the random design. Moreover, the LSE is known to achieve the adaptation rate $(K/n)^{-2/d}\{1\vee \log(n/K)\}^{2\gamma}$ when $f$ is piecewise constant on $K$ hyperrectangles in a partition of $[0,1]^d$.
Due to the minimax theorem, the LSE is identical on every design point to both the max-min and min-max estimators over all upper and lower sets containing the design point. This motivates our consideration of estimators which lie in-between the max-min and min-max estimators over possibly smaller classes of upper and lower sets, including a subclass of block estimators. Under a $q$-th moment condition on the noise, we develop $\ell_q$ risk bounds for such general estimators for isotonic regression on graphs. For uniform deterministic and random designs in $[0,1]^d$ with $d\ge 3$, our $\ell_2$ risk bound for the block estimator matches the minimax rate $n^{-1/d}$ when the range of $f$ is bounded and achieves the near parametric adaptation rate $(K/n)\{1\vee\log(n/K)\}^{d}$ when $f$ is $K$-piecewise constant. Furthermore, the block estimator possesses the following oracle property in variable selection: When $f$ depends on only a subset $S$ of variables, the $\ell_2$ risk of the block estimator automatically achieves up to a poly-logarithmic factor the minimax rate based on the oracular knowledge of $S$.
Submission history
From: Hang Deng [view email][v1] Fri, 21 Dec 2018 04:58:14 UTC (249 KB)
[v2] Fri, 10 Jan 2020 03:39:18 UTC (964 KB)
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