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Biometrika Advance Access published online on June 30, 2009
Biometrika, doi:10.1093/biomet/asp030
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© 2009 Biometrika Trust

Article

Improving point and interval estimators of monotone functions by rearrangement

V. Chernozhukov

Department of Economics, Massachusetts Institute of Technology, 50 Memorial Drive, Cambridge, 02142 Massachusetts, U.S.A. vchern{at}mit.edu

I. Fernández-Val

Department of Economics, Boston University, 270 Bay State Road, Boston, 02215 Massachusetts, U.S.A. ivanf{at}bu.edu

A. Galichon

Ecole Polytechnique, Département d'Economie, 91128 Palaiseau Cedex, France alfred.galichon{at}polytechnique.edu

Received for publication 1 October 2007. Revision received 1 December 2008.
  Abstract

Suppose that a target function is monotonic and an available original estimate of this target function is not monotonic. Rearrangements, univariate and multivariate, transform the original estimate to a monotonic estimate that always lies closer in common metrics to the target function. Furthermore, suppose an original confidence interval, which covers the target function with probability at least 1-{alpha}, is defined by an upper and lower endpoint functions that are not monotonic. Then the rearranged confidence interval, defined by the rearranged upper and lower endpoint functions, is monotonic, shorter in length in common norms than the original interval, and covers the target function with probability at least 1-{alpha}. We illustrate the results with a growth chart example.

Key Words: Growth chart • Improved estimation • Improved inference • Isotonization • Lorentz inequality • Monotone function • Multivariate • Quantile regression • Rearrangement


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