Testing and Estimating Shape-Constrained Nonparametric Density and Regression in the Presence of Measurement Error

Abstract

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 errorfree case.

Fusion of Hard and Soft Information in Nonparametric Density Estimation

“…Although important, we limit the discussion of this topic to Theorems 4.2, 4.4, and 4.7 as well as §5.4; see for example [63] and [9] for tests in related contexts.…”

Making a Non-Parametric Density Estimator More Attractive, and More Accurate, by Data Perturbation

“…Owen's (1988) empirical likelihood method is based on similar ideas, as also are the approaches to density estimation that were suggested by Chen (1997), Zhang (1998), Müller et al (2005) and Schick and Wefelmeyer (2009). Hall and Presnell (1999), Huang (2001, 2002) and Carroll et al (2011) proposed non-parametric function estimators based on tilting constrained by a measure of distance. The use of data sharpening methods is more in its infancy, with contributions made by Braun and Hall (2001) and Müller et al (2003), for example.…”

Parametrically Assisted Nonparametric Estimation of a Density in the Deconvolution Problem

“…As a result, while it can work well when their 5-knot spline model is a good approximation to the true ratio, it can perform significantly worse than standard deconvolution estimators in other cases. Carroll, Delaigle, and Hall (2011) proposed an estimator that includes qualitative constraints (e.g., unimodality), but their method is computationally involved and can offer only modest improvements; for example, it does not improve convergence rates of conventional nonparametric estimators. In this article, we consider estimation of a density f X based on error-contaminated data, especially when f X is known or believed to be close to a parametric family of densities f ( · | θ ) where θ is a vector-valued parameter.…”