Contraction and uniform convergence of isotonic regression

We consider the problem of isotonic regression, where the underlying signal $x$ is assumed to satisfy a monotonicity constraint, that is, $x$ lies in the cone $\{ x\in\mathbb{R}^n : x_1 \leq \dots \leq x_n\}$. We study the isotonic projection operator (projection to this cone), and find a necessary and sufficient condition characterizing all norms with respect to which this projection is contractive. This enables a simple and non-asymptotic analysis of the convergence properties of isotonic regression, yielding uniform confidence bands that adapt to the local Lipschitz properties of the signal.

[1]  S. Geer.  Hellinger-Consistency of Certain Nonparametric Maximum Likelihood Estimators , 1993 .

[2]  Adityanand Guntuboyina,et al.  On risk bounds in isotonic and other shape restricted regression problems , 2013, 1311.3765.

[3]  Caroline J. Klivans,et al.  A Geometric Interpretation of the Characteristic Polynomial of Reflection Arrangements , 2009, 0906.2208.

[4]  T. Gasser,et al.  Residual variance and residual pattern in nonlinear regression , 1986 .

[5]  Hendrik P. Lopuhaa,et al.  The limit distribution of the L∞ -error of Grenander-type estimators , 2011, 1111.5934.

[6]  P. Groeneboom.  Estimating a monotone density , 1984 .

[7]  Prakasa Rao.  Estimation of a unimodal density , 1969 .

[8]  Mary C. Meyer,et al.  ON THE DEGREES OF FREEDOM IN SHAPE-RESTRICTED REGRESSION , 2000 .

[9]  U. Grenander.  On the theory of mortality measurement , 1956 .

[10]  P. Massart,et al.  Rates of convergence for minimum contrast estimators , 1993 .

[11]  Fadoua Balabdaoui,et al.  ON THE GRENANDER ESTIMATOR AT ZERO. , 2009, Statistica Sinica.

[12]  Asymptotic behavior of the grenander estimator at density flat regions , 1999 .

[13]  Cun-Hui Zhang.  Risk bounds in isotonic regression , 2002 .

[14]  Eric Cator,et al.  Adaptivity and optimality of the monotone least-squares estimator , 2008 .

[15]  Shih-Ping Han,et al.  A successive projection method , 1988, Math. Program..

[16]  H. D. Brunk,et al.  Statistical inference under order restrictions : the theory and application of isotonic regression , 1973 .

[17]  F. T. Wright,et al.  Order restricted statistical inference , 1988 .

[18]  J. Rice.  Bandwidth Choice for Nonparametric Regression , 1984 .

[19]  S. Geer.  Estimating a Regression Function , 1990 .

[20]  L. Birge.  Estimating a Density under Order Restrictions: Nonasymptotic Minimax Risk , 1987 .

[21]  F. T. Wright.  The Asymptotic Behavior of Monotone Regression Estimates , 1981 .

[22]  H. Jankowski.  Convergence of linear functionals of the Grenander estimator under misspecification , 2012, 1207.6614.

[23]  Yazhen Wang,et al.  The L2risk of an isotonic estimate , 1996 .

[24]  L. Duembgen.  Optimal Confidence Bands for Shape-Restricted Curves , 2003, 1312.6466.