In many applications we can expect that, or are interested to know if, a density function or a regression curve satisfies some specific shape constraints. For example, when the explanatory variable, X, represents the value taken by a treatment or dosage, the conditional mean of the response, Y , is often anticipated to be a monotone function of X. Indeed, if this regression mean is not monotone (in the appropriate direction) then the medical or commercial value of the treatment is likely to be significantly curtailed, at least for values of X that lie beyond the point at which monotonicity fails. In the case of a density, common shape constraints include log-concavity and unimodality. If we can correctly guess the shape of a curve, then nonparametric estimators can be improved by taking this information into account. Addressing such problems requires a method for testing the hypothesis that the curve of interest satisfies a shape constraint, and, if the conclusion of the test is positive, a technique for estimating the curve subject to the constraint. Nonparametric methodology for solving these problems already exists, but only in cases where the covariates are observed precisely. However in many problems, data can only be observed with measurement errors, and the methods employed in the error-free case typically do not carry over to this error context. In this paper we develop a novel approach to hypothesis testing and function estimation under shape constraints, which is valid in the context of measurement errors. Our method is based on tilting an estimator of the density or the regression mean until it satisfies the shape constraint, and we take as our test statistic the distance through which it is tilted. Bootstrap methods are used to calibrate the test. The constrained curve estimators that we develop are also based on tilting, and in that context our work has points of contact with methodology in the error-free case.
|Original language||English (US)|
|Number of pages||12|
|Journal||Journal of the American Statistical Association|
|State||Published - Mar 2011|
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUS-CI-016-04
Acknowledgements: Raymond J. Carroll is Distinguished Professor, Department of Statistics, Texas A&M University, College Station, TX 77843 (E-mail: email@example.com). Aurore Delaigle is Principal Researcher and Queen Elizabeth II Fellow, Department of Mathematics and Statistics, University of Melbourne, VIC, 3010, Australia (E-mail: A.Delaigle@ms.unimelb.edu.au). Peter Hall is Professor and Federation Fellow, Department of Mathematics and Statistics, University of Melbourne, VIC, 3010, Australia and Distinguished Professor, Department of Statistics, University of California at Davis, Davis, CA 95616 (E-mail: firstname.lastname@example.org). Carroll's research was supported by a grant from the National Cancer Institute (R37-CA057030) and by award number KUS-CI-016-04, made by King Abdullah University of Science and Technology (KAUST), and by National Science Foundation Instrumentation grant number 0922866. Delaigle's research was supported by grants from the Australian Research Council, and Hall's research was supported by grants from the Australian Research Council and from the National Science Foundation. The authors thank the editor, the associate editor, and a referee for their valuable comments that helped improve a previous version of the manuscript.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.