Improving Point and Interval Estimates of Monotone Functions by Rearrangement

31 Pages Posted: 16 Jul 2008  

Victor Chernozhukov

Massachusetts Institute of Technology (MIT) - Department of Economics; New Economic School

Iván Fernández‐Val

Boston University - Department of Economics

Alfred Galichon

NYU, Department of Economics and Courant Institute

Date Written: July 14, 2008

Abstract

Suppose that a target function is monotonic, namely, weakly increasing, and an original estimate of this target function is available, which is not weakly increasing. Many common estimation methods used in statistics produce such estimates. We show that these estimates can always be improved with no harm using rearrangement techniques: The rearrangement methods, univariate and multivariate, transform the original estimate to a monotonic estimate, and the resulting estimate is closer to the true curve in common metrics than the original estimate. The improvement property of the rearrangement also extends to the construction of confidence bands for monotone functions. Suppose we have the lower and upper endpoint functions of a simultaneous confidence interval that covers the target function with a pre-specified probability level, then the rearranged confidence interval, defined by the rearranged lower and upper end-point functions, is shorter in length in common norms than the original interval and covers the target function with probability greater or equal to the pre-specified level. We illustrate the results with a computational example and an empirical example dealing with age-height growth charts.

Keywords: Monotone function, improved estimation, improved inference, multivariate rearrangement, univariate rearrangement, Lorentz inequalities, growth chart, quantile regression, mean regression, series, locally linear, kernel methods

JEL Classification: 62G08, 46F10, 62F35, 62P10

Suggested Citation

Chernozhukov, Victor and Fernández‐Val, Iván and Galichon, Alfred, Improving Point and Interval Estimates of Monotone Functions by Rearrangement (July 14, 2008). MIT Department of Economics Working Paper No. 08-13. Available at SSRN: https://ssrn.com/abstract=1159965 or http://dx.doi.org/10.2139/ssrn.1159965

Victor Chernozhukov (Contact Author)

Massachusetts Institute of Technology (MIT) - Department of Economics ( email )

50 Memorial Drive
Room E52-262f
Cambridge, MA 02142
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617-253-4767 (Phone)
617-253-1330 (Fax)

HOME PAGE: http://www.mit.edu/~vchern/

New Economic School

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Moscow, 117418
Russia

Ivan Fernandez-Val

Boston University - Department of Economics ( email )

270 Bay State Road
Boston, MA 02215
United States

HOME PAGE: http://people.mit.edu/ivanf

Alfred Galichon

NYU, Department of Economics and Courant Institute ( email )

269 Mercer Street, 7th Floor
New York, NY 10011
United States

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