Abstract

The venerable method of maximum likelihood has found numerous recent applications in nonparametric estimation of regression and shape constrained densities. For mixture models the nonparametric maximum likelihood estimator (NPMLE) of Kiefer and Wolfowitz plays a central role in recent developments of empirical Bayes methods. The NPMLE has also been proposed by Cosslett as an estimation method for single index linear models for binary response with random coefficients. However, computational difficulties have hindered its application. Combining recent developments in computational geometry and convex optimization, we develop a new approach to computation for such models that dramatically increases their computational tractability. Consistency of the method is established for an expanded profile likelihood formulation. The methods are evaluated in simulation experiments, compared to the deconvolution methods of Gautier and Kitamura and illustrated in an application to modal choice for journey-to-work data in the Washington DC area. Supplementary materials for this article are available online.

摘要

古老的最大似然方法最近在非参数中发现了许多应用回归和形状约束密度的估计。对于混合模型，Kiefer 和 Wolfowitz 的非参数最大似然估计器 (NPMLE) 在经验贝叶斯方法的最新发展中发挥着核心作用。Cosslett 还提出了 NPMLE 作为具有随机系数的二元响应的单指数线性模型的估计方法。然而，计算困难阻碍了它的应用。结合计算几何和凸优化的最新发展，我们为此类模型开发了一种新的计算方法，显着提高了它们的计算易处理性。该方法的一致性是为扩展的轮廓似然公式建立的。这些方法在模拟实验中进行评估，与 Gautier 和 Kitamura 的反卷积方法相比，并在华盛顿特区地区上班途中数据模态选择的应用中进行了说明。本文的补充材料可在线获取。