论文标题
关于在回归模型中使用$ l $功能
On the Use of $L$-functionals in Regression Models
论文作者
论文摘要
在本文中，我们调查并统一回归模型中响应变量条件分布的大型或$ L $函数。这包括在协变量条件下的位置，尺度，偏度和响应量的重量量的强大度量。我们概括了$ l $ -Moments（Sittinen，1969），$ L $ -Skewness和$ L $ -Kurtosis（Hosking，1990），并通过正交系列量化功能扩展。特别是，我们激励着为什么位置，规模，偏度和繁重的订单分别具有1、2，（3,2）和（4,2）的订单，并描述了$ l $ unctionals（具有不同订单数字的家庭）如何由Legendre，Hermite，Hermite，Laguerre，Laguerre或其他类型的多面年来构建。我们的框架应用于模型，在该模型中，响应的分位数与协变量之间的关系遵循转换的线性模型，其链路函数确定了适当的$ L $ functionals类。在这种情况下，对响应的分布进行参数或非参数处理，并且对响应变量进行审查/截断。我们还提供了$ l $ unctionals估计的统一渐近理论，并通过分析迁移鸟类的到达时间分布来说明我们的方法。在这种情况下，引入了一种新颖的确定系数，它利用了上述正交系列扩展。
In this paper we survey and unify a large class or $L$-functionals of the conditional distribution of the response variable in regression models. This includes robust measures of location, scale, skewness, and heavytailedness of the response, conditionally on covariates. We generalize the concepts of $L$-moments (Sittinen, 1969), $L$-skewness, and $L$-kurtosis (Hosking, 1990) and introduce order numbers for a large class of $L$-functionals through orthogonal series expansions of quantile functions. In particular, we motivate why location, scale, skewness, and heavytailedness have order numbers 1, 2, (3,2), and (4,2) respectively and describe how a family of $L$-functionals, with different order numbers, is constructed from Legendre, Hermite, Laguerre or other types of polynomials. Our framework is applied to models where the relationship between quantiles of the response and the covariates follow a transformed linear model, with a link function that determines the appropriate class of $L$-functionals. In this setting, the distribution of the response is treated parametrically or nonparametrically, and the response variable is either censored/truncated or not. We also provide a unified asymptotic theory of estimates of $L$-functionals, and illustrate our approach by analyzing the arrival time distribution of migrating birds. In this context a novel version of the coefficient of determination is introduced, which makes use of the abovementioned orthogonal series expansion.
