On risk bounds in isotonic and other shape restricted regression problems

Sabyasachi Chatterjee, Adityanand Guntuboyina, Bodhisattva Sen

Research output: Contribution to journalArticlepeer-review


We consider the problem of estimating an unknown θ ∈ ℝn from noisy observations under the constraint that θ belongs to certain convex polyhedral cones in ℝn. Under this setting, we prove bounds for the risk of the least squares estimator (LSE). The obtained risk bound behaves differently depending on the true sequence θ which highlights the adaptive behavior of θ. As special cases of our general result, we derive risk bounds for the LSE in univariate isotonic and convex regression. We study the risk bound in isotonic regression in greater detail: we show that the isotonic LSE converges at a whole range of rates from log n/n (when θ is constant) to n-2/3 (when θ is uniformly increasing in a certain sense). We argue that the bound presents a benchmark for the risk of any estimator in isotonic regression by proving nonasymptotic local minimax lower bounds. We prove an analogue of our bound for model misspecification where the true θ is not necessarily nondecreasing.

Original languageEnglish (US)
Pages (from-to)1774-1800
Number of pages27
JournalAnnals of Statistics
Issue number4
StatePublished - Aug 1 2015
Externally publishedYes


  • Adaptation
  • Convex polyhedral cones
  • Global risk bounds
  • Local minimax bounds
  • Model misspecification
  • Statistical dimension

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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