Shape-Constrained Inference for Concave-Transformed Densities and their Modes
Shape-Constrained Inference for Concave-Transformed Densities and their Modes
复制标题
凹变换密度及其模式的形状约束推理
DOI：
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
Charles R. Doss
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文献类型：
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作者：
Charles R. Doss
凹变换密度及其模式的形状约束推断。Doss监事会主席：Jon A. Wellner统计我们考虑通过形状约束估计的函数的推断基于？我们考虑对数凹密度和其他“凹变换”密度的真实的线，其中一个凹变换类是一个通过应用变换（例如对数或幂函数）凹函数。我们希望我们的证明和结果在其他基于凹性的设置中是相关的。凹函数总是单峰的，因此凹变换密度可以用作单峰密度的替代，因此模式是感兴趣的自然参数。在非参数设置的模式通常是不可估计的根-n率，并不总是有一个正常的极限分布，目前的方法测试或形成置信区间的位置模式通常是复杂的。在对数凹密度估计的背景下，通过比较对数凹极大似然估计和对数凹密度约束子集上的固定模极大似然估计，构造了模位置的似然比检验.该测试可以反向以形成置信集。研究了约束极大似然估计的性质和似然比统计量的Wilks现象。证明全局收敛率的n2/5，约束和无约束的MLE，是理解似然比统计的重要一步，这一结果也是独立的利益。这些全球率的结果适用于海灵格和总变差距离，以及似然比统计量的大小，他们适用于许多凹变换的密度类超出对数凹的。
Shape-Constrained Inference for Concave-Transformed Densities and their Modes Charles R. Doss Chair of the Supervisory Committee: Professor Jon A. Wellner Statistics We consider inference about functions estimated via shape constraints based on concavity. We consider log-concave densities and other “concave-transformed” densities on the real line, where a concave-transformed class is one given by applying a transformation (e.g. the logarithm or a power function) to concave functions. We expect our proofs and results to be relevant in other concavity-based settings. Concave functions are always unimodal, so concave-transformed densities can be used as surrogates for unimodal ones, and the mode is thus a natural parameter of interest. In nonparametric settings the mode is generally not estimable at a root-n rate and does not always have a normal limiting distribution, and current methods for testing or forming confidence intervals for the location of the mode are generally complicated. In the setting of log-concave density estimation we construct a likelihood ratio test for the location of the mode by comparing the log-concave maximum likelihood estimate (MLE) to the MLE over the constrained subclass of log-concave densities with a fixed mode. The test can be inverted to form a confidence set. We study the properties of the constrained MLE and the Wilks phenomenon of the likelihood ratio statistic. Proving global rates of convergence of n2/5, for both the constrained and unconstrained MLEs, is an important step in understanding the likelihood ratio statistic and this result is also of independent interest. These global rate results apply to Hellinger and total variation distance, as well as to the size of the likelihood ratio statistic, and they apply to many concave-transformed density classes beyond log-concave ones.
影响因子：
4.5
作者：
Seregin,Arseni;Wellner,JonA
通讯作者：
Wellner,JonA