Abstract

A number of tests have been proposed for assessing the location-scale assumption that is often invoked by practitioners. Existing approaches include Kolmogorov–Smirnov and Cramer–von Mises statistics that each involve measures of divergence between unknown joint distribution functions and products of marginal distributions. In practice, the unknown distribution functions embedded in these statistics are typically approximated using nonsmooth empirical distribution functions (EDFs). In a recent article, Li, Li, and Racine establish the benefits of smoothing the EDF for inference, though their theoretical results are limited to the case where the covariates are observed and the distributions unobserved, while in the current setting some covariates and their distributions are unobserved (i.e., the test relies on population error terms from a location-scale model) which necessarily involves a separate theoretical approach. We demonstrate how replacing the nonsmooth distributions of unobservables with their kernel-smoothed sample counterparts can lead to substantial power improvements, and extend existing approaches to the smooth multivariate and mixed continuous and discrete data setting in the presence of unobservables. Theoretical underpinnings are provided, Monte Carlo simulations are undertaken to assess finite-sample performance, and illustrative applications are provided.

摘要

已经提出了许多用于评估从业人员经常调用的位置比例假设的测试。现有方法包括Kolmogorov-Smirnov和Cramer-von Mises统计，每种统计方法都涉及未知联合分布函数和边际分布乘积之间的差异度量。实际上，嵌入这些统计信息中的未知分布函数通常使用非平滑的经验分布函数（EDF）进行近似。在最近的一篇文章，李，李，和拉辛建立平滑的推论EDF的好处，但他们的理论结果仅限于在协变量的情况下观察和分布不可观测，而在当前设置一些协变量和它们的分布是未观察到的（即，测试依赖于位置比例模型中的人口误差项），这必然涉及单独的理论方法。我们演示了如何用内核平滑的样本对应物替换不可观测变量的不平滑分布可以导致显着的功效提升，并在存在不可观测变量的情况下将现有方法扩展到平滑的多元变量以及混合的连续和离散数据设置。提供理论基础，进行蒙特卡洛模拟以评估有限样本性能，并提供说明性应用。