Massachusetts Institute of Technology Department of Economics Working Paper Series Improving Point and Interval Estimates of Monotone Functions by Rearrangement Improving Point and Interval Estimates of Monotone Functions by Rearrangement

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 by 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. Let l and u be the lower and upper endpoint functions of a simultaneous confidence interval [l,u] that covers the true function with probability (1-a), 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 true function with probability greater or equal to (1-a). We illustrate the results with a computational example and an empirical example dealing with age-height growth charts. Please note: This paper is a revised version of cemmap working Paper CWP09/07.

[1]  George G. Lorentz,et al.  An Inequality for Rearrangements , 1953 .

[2]  H. D. Brunk,et al.  AN EMPIRICAL DISTRIBUTION FUNCTION FOR SAMPLING WITH INCOMPLETE INFORMATION , 1955 .

[3]  E. Lehmann.  Testing Statistical Hypotheses , 1960 .

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

[5]  J. Kalbfleisch.  Statistical Inference Under Order Restrictions , 1975 .

[6]  A. Gallant,et al.  On the bias in flexible functional forms and an essentially unbiased form : The fourier flexible form , 1981 .

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

[8]  Tj Cole,et al.  Fitting smoothed centile curves to reference data (with discussion) , 1988 .

[9]  T. Cole.  Fitting Smoothed Centile Curves to Reference Data , 1988 .

[10]  J. Ramsay.  Monotone Regression Splines in Action , 1988 .

[11]  Iain M. Johnstone,et al.  Hotelling's Theorem on the Volume of Tubes: Some Illustrations in Simultaneous Inference and Data Analysis , 1990 .

[12]  Probal Chaudhuri,et al.  Nonparametric Estimates of Regression Quantiles and Their Local Bahadur Representation , 1991 .

[13]  D. Andrews.  Asymptotic Normality of Series Estimators for Nonparametric and Semiparametric Regression Models , 1991 .

[14]  E. Mammen.  Nonparametric regression under qualitative smoothness assumptions , 1991 .

[15]  P. Hall.  On Edgeworth Expansion and Bootstrap Confidence Bands in Nonparametric Curve Estimation , 1993 .

[16]  Jianqing Fan,et al.  Local polynomial modelling and its applications , 1994 .

[17]  C. J. Stone,et al.  The Use of Polynomial Splines and Their Tensor Products in Multivariate Function Estimation , 1994 .

[18]  Rosa L. Matzkin.  Restrictions of economic theory in nonparametric methods , 1994 .

[19]  Matthew P. Wand,et al.  Kernel Smoothing , 1995 .

[20]  Anne-Laure Fougères,et al.  Estimation de densités unimodales , 1997 .

[21]  W. Newey,et al.  Convergence rates and asymptotic normality for series estimators , 1997 .

[22]  Stephen Portnoy,et al.  Local asymptotics for quantile smoothing splines , 1997 .

[23]  J. Ramsay.  Estimating smooth monotone functions , 1998 .

[24]  Q. Shao,et al.  On Parameters of Increasing Dimensions , 2000 .

[25]  E. Mammen,et al.  A General Projection Framework for Constrained Smoothing , 2001 .

[26]  D. Pollard.  A User's Guide to Measure Theoretic Probability by David Pollard , 2001 .

[27]  K. Knight.  What are the Limiting Distributions of Quantile Estimators , 2002 .

[28]  C. Villani.  Topics in Optimal Transportation , 2003 .

[29]  G. Burton.  TOPICS IN OPTIMAL TRANSPORTATION (Graduate Studies in Mathematics 58) By CÉDRIC VILLANI: 370 pp., US$59.00, ISBN 0-8218-3312-X (American Mathematical Society, Providence, RI, 2003) , 2004 .

[30]  Holger Dette,et al.  A comparative study of monotone nonparametric kernel estimates , 2006 .

[31]  Ričardas Zitikis,et al.  An index of monotonicity and its estimation: a step beyond econometric applications of the Gini index , 2005 .

[32]  H. Dette,et al.  Strictly monotone and smooth nonparametric regression for two or more variables , 2005 .

[33]  Roger Koenker,et al.  Inequality constrained quantile regression , 2005 .

[34]  L. Wasserman.  All of Nonparametric Statistics , 2005 .

[35]  Holger Dette,et al.  Strictly monotone and smooth nonparametric regression for two or more variables , 2005 .

[36]  H. Dette,et al.  A simple nonparametric estimator of a strictly monotone regression function , 2006 .

[37]  Larry Wasserman,et al.  All of Nonparametric Statistics (Springer Texts in Statistics) , 2006 .

[38]  James O. Ramsay,et al.  Functional Data Analysis , 2005 .

[39]  R. Koenker,et al.  Quantile regression methods for reference growth charts , 2006, Statistics in medicine.

[40]  Stergios B. Fotopoulos,et al.  All of Nonparametric Statistics , 2007, Technometrics.

[41]  Susan A. Murphy,et al.  Monographs on statistics and applied probability , 1990 .

[42]  Victor Chernozhukov,et al.  Improving Estimates of Monotone Functions by Rearrangement , 2007, 0704.3686.

[43]  Victor Chernozhukov,et al.  Rearranging Edgeworth–Cornish–Fisher expansions , 2007, 0708.1627.

[44]  V. Chernozhukov,et al.  QUANTILE AND PROBABILITY CURVES WITHOUT CROSSING , 2007, 0704.3649.

[45]  R. Koenker,et al.  Regression Quantiles , 2007 .

[46]  Christopher R. Genovese,et al.  Adaptive confidence bands , 2007 .