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Biometrika Advance Access originally published online on June 30, 2009
Biometrika 2009 96(3):559-575; 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.

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



References

    Andrews D. W. K. Asymptotic normality of series estimators for nonparametric and semiparametric regression models. Econometrica (1991) 59:307–45.[CrossRef][Web of Science]

    Ayer M., Brunk H. D., Ewing G. M., Reid W. T., Silverman E. An empirical distribution function for sampling with incomplete information. Ann. Math. Statist. (1955) 26:641–7.[CrossRef]

    Barlow R. E., Bartholomew D. J., Bremner J. M., Brunk H. D. Statistical Inference under Order Restrictions. The Theory and Application of Isotonic Regression (1972) New York: John Wiley & Sons.

    Chaudhuri P. Nonparametric estimates of regression quantiles and their local Bahadur representation. Ann. Statist. (1991) 19:760–77.[CrossRef]

    Chernozhukov V., Fernández-Val I., Galichon A. Rearranging Edgeworth–Cornish–Fisher expansions. Econ. Theory (2009) doi: 10.1007/s00199-008-0431-z.

    Cole T. J. Fitting smoothed centile curves to reference data. J. R. Statist. Soc. A (1988) 151:385–418.[CrossRef]

    Davydov Y., Zitikis R. An index of monotonicity and its estimation: a step beyond econometric applications of the Gini index. Metron. (2005) 63:351–72.

    Dette H., Neumeyer N., Pilz K. F. A simple nonparametric estimator of a strictly monotone regression function. Bernoulli (2006) 12:469–90.[CrossRef][Web of Science]

    Dette H., Scheder R. Strictly monotone and smooth nonparametric regression for two or more variables. Can. J. Statist. (2006) 34:535–61.[CrossRef]

    Fan J., Gijbels I. Monographs on Statistics and Applied Probability. In: Local Polynomial Modelling and Its Applications (1996) 66. London: Chapman & Hall.

    Fougeres A.-L. Estimation de densites unimodales. Can. J. Statist. (1997) 25:375–87.[CrossRef]

    Gallant A. R. On the bias in flexible functional forms and an essentially unbiased form: the Fourier flexible form. J. Economet. (1981) 15:211–45.[CrossRef]

    Genovese C., Wasserman L. Adaptive confidence bands. Ann. Statist. (2008) 36:875–905.[CrossRef]

    Hall P. On Edgeworth expansion and bootstrap confidence bands in nonparametric curve estimation. J. R. Statist. Soc. B (1993) 55:291–304.

    Hardy G. H., Littlewood J. E., Pólya G. Inequalities (1952) 2nd ed. Cambridge: Cambridge University Press.

    He X., Shao Q.-M. On parameters of increasing dimensions. J. Mult. Anal. (2000) 73:120–35.[CrossRef]

    Koenker R., Bassett G. S. Regression quantiles. Econometrica (1978) 46:33–50.[CrossRef][Web of Science]

    Koenker R., Ng P. Inequality constrained quantile regression. Sankhya (2005) 67:418–40.

    Lehmann E. L., Romano J. P. Testing Statistical Hypotheses (2005) 3rd ed. New York: Springer.

    Lorentz G. G. An inequality for rearrangements. Am. Math. Mon. (1953) 60:176–9.[CrossRef]

    Mammen E. Nonparametric regression under qualitative smoothness assumptions. Ann. Statist. (1991) 19:741–59.[CrossRef]

    Mammen E., Marron J. S., Turlach B. A., Wand M. P. A general projection framework for constrained smoothing. Statist. Sci. (2001) 16:232–48.[CrossRef]

    Matzkin R. L. Restrictions of economic theory in nonparametric methods. Handbook of Econometrics—Engle R. F., McFadden D. L., eds. (1994) 4. Amsterdam: North-Holland. 2523–58.[CrossRef]

    Newey W. K. Convergence rates and asymptotic normality for series estimators. J. Economet. (1997) 79:147–68.[CrossRef]

    Pollard D. Cambridge Series in Statistical and Probabilistic Mathematics. In: A User's Guide to Measure Theoretic Probability (2002) 8. Cambridge: Cambridge University Press.

    Portnoy S. Local asymptotics for quantile smoothing splines. Ann. Statist. (1997) 25:414–34.[CrossRef]

    Ramsay J. O. Monotone regression splines in action. Statist. Sci. (1988) 3:425–41.[CrossRef]

    Ramsay J. O. Estimating smooth monotone functions. J. R. Statist. Soc. B (1998) 60:365–75.[CrossRef]

    Ramsay J. O., Silverman B. W. Functional Data Analysis (2005) 2nd ed. New York: Springer.

    Robertson T., Wright F. T., Dykstra R. L. Order Restricted Statistical Inference (1988) Chichester: John Wiley & Sons.

    Silvapulle M. J., Sen P. K. Constrained Statistical Inference (2005) Hoboken, NJ: John Wiley & Sons.

    Stone C. J. The use of polynomial splines and their tensor products in multivariate function estimation. Ann. Statist. (1994) 22:118–84.[CrossRef]

    van der Vaart A. W. Asymptotic Statistics (1998) Cambridge: Cambridge University Press.

    Villani C. Graduate Studies in Mathematics. In: Topics in Optimal Transportation (2003) 58. Providence, RI: American Mathematical Society.

    Wand M. P., Jones M. C. Kernel Smoothing. (1995) London: Chapman and Hall.

    Wasserman L. All of Nonparametric Statistics (2006) New York: Springer.

    Wei Y., Pere A., Koenker R., He X. Quantile regression methods for reference growth charts. Statist. Med. (2006) 25:1369–82.[CrossRef]


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