Iterative isotonic regression

Guyader, Arnaud; Hengartner, Nick; Jégou, Nicolas; Matzner-Løber, Eric

ESAIM: Probability and Statistics, Tome 19 (2015), pp. 1-23

This article explores some theoretical aspects of a recent nonparametric method for estimating a univariate regression function of bounded variation. The method exploits the Jordan decomposition which states that a function of bounded variation can be decomposed as the sum of a non-decreasing function and a non-increasing function. This suggests combining the backfitting algorithm for estimating additive functions with isotonic regression for estimating monotone functions. The resulting iterative algorithm is called Iterative Isotonic Regression (I.I.R.). The main result in this paper states that the estimator is consistent if the number of iterations $k_n$ grows appropriately with the sample size $n$. The proof requires two auxiliary results that are of interest in and by themselves: firstly, we generalize the well-known consistency property of isotonic regression to the framework of a non-monotone regression function, and secondly, we relate the backfitting algorithm to von Neumann’s algorithm in convex analysis. We also analyse how the algorithm can be stopped in practice using a data-splitting procedure.

Classification : 52A05, 62G08, 62G20
Keywords: Nonparametric statistics, isotonic regression, additive models, metric projection onto convex cones

D. Anevski and P. Soulier, Monotone spectral density estimation. Ann. Stat. 39 (2011) 418–438. | MR | Zbl

M. Ayer, H.D. Brunk, G.M. Ewing, W.T. Reid and E. Silverman, An empirical distribution function for sampling with incomplete information. Ann. Math. Stat. (1955) 641–647. | MR | Zbl

R.E. Barlow, D.J. Bartholomew, J.M. Bremner and H.D. Brunk, Statistical inference under order restrictions: Theory and application of isotonic regression. John Wiley & Sons (1972). | MR | Zbl

H.H. Bauschke and J.M. Borwein, On the Convergence of von Neumann’s Alternating Projection Algorithm for Two Sets. Set-Valued Anal. 1 (1993) 185–212. | MR | Zbl

H.H. Bauschke and J.M. Borwein, Dykstra’s alternating projection algorithm for two sets. J. Approx. Theory 79 (1994) 418–443. | MR | Zbl

M.J. Best and N. Chakravarti, Active set algorithms for isotonic regression; An unifying framework. Math. Program. 47 (1990) 425–439. | MR | Zbl

H.D. Brunk, Estimation of isotonic regression. Cambridge University Press (1970) 177–195. | MR

H.D. Brunk, Maximum likelihood estimates of monotone parameters. Ann. Math. Stat. (1955) 607–616. | MR | Zbl

V.V. Buldygin and Y.V. Kozachenko, Metric Characterization of Random Variables and Random Processes. American Mathematical Society (1972). | MR | Zbl

A. Buja, T.J. Hastie and R.J. Tibshirani, Linear smoothers and additive models. Ann. Stat. 17 (1989) 453–510. | MR | Zbl

F. Deutsch, The method of alternating orthogonal projections. Approximation Theory, Spline Functions and Applications, edited by S.P. Singh (1991) 105–121. | MR | Zbl

C. Durot, On the Lp-error of monotonicity constrained estimators. Ann. Stat. 35 (2007) 1080–1104. | MR | Zbl

R.L. Dykstra, An isotonic regression algorithm. J. Stat. Plann. Inference 5 (1981) 355–363. | MR | Zbl

J.H. Friedman and W. Stuetzle, Projection pursuit regression. J. Amer. Stat. Assoc. (1981) 817–823. | MR

A. Guyader, N. Jégou, A.B. Németh and S.N. Németh, A Geometrical Approach to Iterative Isotone Regression. Appl. Math. Comput. 227 (2014) 359–369. | MR | Zbl

L. Györfi, M. Kohler, A. Kryzak and H. Walk, A distribution-free theory of nonparametric regression. Springer-Verlag, New York (1990). | MR | Zbl

D.L. Hanson, G. Pledger and F.T. Wright, On consistency in monotonic regression. Ann. Stat. 1 (1973) 401–421. | MR | Zbl

T.J. Hastie and R.J. Tibshirani, Generalized additive models. Chapman & Hall/CRC (1990). | MR | Zbl

W. Härdle and P. Hall, On the backfitting algorithm for additive regression models. Statistica Neerlandica 47 (1993) 43–57. | MR | Zbl

N.W. Hengartner and S. Sperlich, Rate optimal estimation with the integration method in the presence of many covariates. J. Multivar. Anal. 95 (1999) 246–272. | MR | Zbl

J. Horowitz, J. Klemelä and E. Mammen, Optimal estimation in additive regression models. Bernoulli 12 (2006) 271–298. | MR | Zbl

W. Kim, O.B. Linton and N.W. Hengartner, A computationally efficient oracle estimator for additive nonparametric regression with bootstrap confidence intervals. J. Comput. Graph. Stat. 8 (1999) 278–297. | MR

C.I.C. Lee, The min-max algorithm and isotonic regression. Ann. Stat. 11 (1983) 467–477. | MR | Zbl

E. Mammen and K. Yu, Additive isotone regression. In: Asymptotics: Particles, Processes and Inverse Problems, Lect. Notes Monogr. Series 55 (2007) 179–195. | MR | Zbl

E. Mammen, O. Linton and J. Nielsen, The existence and asymptotic properties of a backfitting projection algorithm under weak conditions. Ann. Stat. 27 (1999) 1443–1490. | MR | Zbl

M. Meyer and M. Woodroofe, On the Degrees of Freedom in Shape-Restricted Regression. Ann. Stat. 28 (2000) 1083–1104. | MR | Zbl

J.D. Opsomer, Asymptotic properties of backfitting estimators. J. Multivar. Anal. 73 (2000) 166–179. | MR | Zbl

J.D. Opsomer and D. Ruppert, Fitting a bivariate additive model by local polynomial regression. Ann. Stat. 25 (1997) 186–211. | MR | Zbl

T. Robertson, F.T. Wright and R.L. Dykstra, Order Restricted Statistical Inference. Wiley, New York (1988). | MR | Zbl

S. Van De Geer, and M. Wegkamp, Consistency for the least squares estimator in nonparametric regression. Ann. Stat. 24 (1996) 2513–2523. | MR | Zbl

S. van de Geer, Empirical Process in M-Estimation. Cambridge University Press (2000). | Zbl