Details

Autor(en) / Beteiligte

• Deng, Hang
• Zhang, Cun-Hui

Titel

ISOTONIC REGRESSION IN MULTI-DIMENSIONAL SPACES AND GRAPHS

Ist Teil von

• The Annals of statistics, 2020-12, Vol.48 (6), p.3672-3698

Ort / Verlag

Hayward: Institute of Mathematical Statistics

Erscheinungsjahr

2020

Quelle

Project Euclid Complete

Beschreibungen/Notizen

• 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 ℓ₂ 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) γ where n is the sample size, γ = 4 in the lattice design and γ = max{9/2, (d² + d + 1)/2} in the random design. Moreover, the LSE is known to achieve the adaptation rate (K/n)−2/d {1 ∨ log(n/K)}2γ 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 ℓ₂ 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 ∨ 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 ℓ₂ risk of the block estimator automatically achieves up to a poly-logarithmic factor the minimax rate based on the oracular knowledge of S.