Carroll Raymond J, Delaigle Aurore, Hall Peter
Department of Statistics, Texas A & M University, College Station, TX 77843 (
J Am Stat Assoc. 2011 Mar;106(493):191-202. doi: 10.1198/jasa.2011.tm10355.
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.
在许多应用中，我们可以预期，或者有兴趣了解一个密度函数或回归曲线是否满足某些特定的形状约束。例如，当解释变量X表示治疗或剂量的值时，响应变量Y的条件均值通常被预期为X的单调函数。事实上，如果这个回归均值不是单调的（在适当的方向上），那么治疗的医学或商业价值可能会显著降低，至少对于X值超出单调性失效点的情况是如此。对于密度而言，常见的形状约束包括对数凹性和单峰性。如果我们能够正确猜测曲线的形状，那么非参数估计量可以通过考虑这些信息而得到改进。解决此类问题需要一种方法来检验感兴趣的曲线满足形状约束这一假设，并且，如果检验结论是肯定的，还需要一种在约束条件下估计曲线的技术。解决这些问题的非参数方法已经存在，但仅适用于协变量被精确观测的情况。然而，在许多问题中，数据只能带有测量误差进行观测，并且在无误差情况下所采用的方法通常不能直接应用于此误差情形。在本文中，我们开发了一种在形状约束下进行假设检验和函数估计的新方法，该方法在测量误差的背景下是有效的。我们的方法基于对密度或回归均值的估计量进行倾斜，直到它满足形状约束，并且我们将其倾斜的距离作为检验统计量。使用自助法来校准检验。我们开发的约束曲线估计量也基于倾斜，在这种情况下，我们的工作与无误差情况下的方法有联系点。
