Annals of Statistics

Confidence sets for split points in decision trees

Moulinath Banerjee and Ian W. McKeague

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We investigate the problem of finding confidence sets for split points in decision trees (CART). Our main results establish the asymptotic distribution of the least squares estimators and some associated residual sum of squares statistics in a binary decision tree approximation to a smooth regression curve. Cube-root asymptotics with nonnormal limit distributions are involved. We study various confidence sets for the split point, one calibrated using the subsampling bootstrap, and others calibrated using plug-in estimates of some nuisance parameters. The performance of the confidence sets is assessed in a simulation study. A motivation for developing such confidence sets comes from the problem of phosphorus pollution in the Everglades. Ecologists have suggested that split points provide a phosphorus threshold at which biological imbalance occurs, and the lower endpoint of the confidence set may be interpreted as a level that is protective of the ecosystem. This is illustrated using data from a Duke University Wetlands Center phosphorus dosing study in the Everglades.

Article information

Ann. Statist., Volume 35, Number 2 (2007), 543-574.

First available in Project Euclid: 5 July 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G08: Nonparametric regression 62G20: Asymptotic properties 62E20: Asymptotic distribution theory

CART change-point estimation cube-root asymptotics empirical processes logistic regression Poisson regression nonparametric regression split point


Banerjee, Moulinath; McKeague, Ian W. Confidence sets for split points in decision trees. Ann. Statist. 35 (2007), no. 2, 543--574. doi:10.1214/009053606000001415.

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