ISOTONIC REGRESSION IN MULTI-DIMENSIONAL SPACES AND GRAPHS

• 成果类型: Article
• 署名作者: Deng, Hang; Zhang, Cun-Hui
• 署名单位: Rutgers University System; Rutgers University New Brunswick
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/20-AOS1947
• 发表日期: 2020
• 页码: 3672-3698
• 关键词: convergence; RISK

摘要

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 >= 2 and N(0, 1) noise, the minimax rate for the l(2) risk is known to be bounded from below by n(-1/d) when the unknown mean function f is non-decreasing 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 the 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 boolean OR 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 qth moment condition on the noise, we develop l(q) risk bounds for such general estimators for isotonic regression on graphs. For uniform deterministic and random designs in [0, 1](d) with d >= 3, our l(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 boolean OR 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 l(2) risk of the block estimator automatically achieves up to a poly-logarithmic factor the minimax rate based on the oracular knowledge of S.