Flexible density regression methods

该项目的目标是开发一个统一的框架，灵活的半参数回归方法的密度，以更好地描述和理解感兴趣的变量之间的关系。虽然在传统的（标量）数据的平均导向的参数建模的局限性，促进了各种扩展到其他分布特征（分位数）或多个分布参数（分布回归），第一种方法最近已经开发的概率密度作为统计分析的对象的功能和成分数据分析。然而，这些分支的回归与灵活的分布（个人水平的方法）和分布的统计分析（密度水平的方法）迄今已独立开发。我们的使命是与他们一道，富有成效地相互充实，促进方法的发展。为了说明灵活的密度回归方法的需求，我们参考了性别经济学的一个例子：女性在夫妻劳动收入中所占份额的分配对于性别认同规范的问题很重要。作为一个概率分布，它在[0，1]上呈现一个混合分布，在0和1处具有正的概率质量（对于单收入夫妇），在两者之间具有连续的-通常是双峰的-密度（对于双收入夫妇）。然后，将这一分布与可能影响家庭、年份或居住地区中的子女（年龄）等分布的变量联系起来是有意义的。例如，在德国的东部和西部各州之间就存在着明显的差异。密度是特别感兴趣的，因为它使转移的概率质量或双峰由于分组很容易看到，因为它很好地扩展到离散，连续，混合或双变量分布。这种分析需要能够根据协变量灵活地对整个分布建模的方法，而无需正态性等参数假设，并且允许线性，非线性和随机效应。我们将通过开发合适的密度回归方法来解决这个问题。根据数据情况，需要密度回归方法，并将为密度值数据开发密度回归方法-例如，当为管理数据提供直方图或用于汇总大量数据时-以及当兴趣在于对给定协变量的条件密度进行建模时，为单个标量数据开发密度回归方法。在我们的统一方法中，这两个场景呈现了同一枚硬币的两面，而不是引用两个不同的统计分支。我们将开发连续密度（例如收入），离散密度，即组成数据（例如离散类别上的时间使用）以及混合密度（例如妇女在家庭劳动收入中的比例）。此外，我们将把这一点扩展到双变量密度，例如，当共同观察一个家庭中的一对夫妇时，允许克服限制性的相关性假设。

The goal of this project is to develop a unified framework of flexible semiparametric regression methods for densities to better describe and understand relationships between variables of interest. While limitations in traditional mean-oriented parametric modeling of (scalar) data have promoted a variety of extensions to other distributional characteristics (quantiles) or multiple distributional parameters (distributional regression), first methods have recently been developed in functional and compositional data analysis for probability densities as objects of statistical analysis. However, these branches of regression with flexible distributions (individual-level approaches) and statistical analysis of distributions (density-level approaches) have so far been independently developed. Our mission is to join them for fruitful mutual enrichment, facilitating methodological developments. To illustrate the demand of flexible density regression methods, we refer to an example from gender economics: the distribution of the woman's share of a couple's labor income is important for questions on gender identity norms. As a probability distribution, it presents a mixed distribution on [0,1] with positive probability mass at 0 and 1 (for single-income couples) and a continuous - often bimodal - density in between (for double-income couples). It is then of interest to relate this distribution to variables that may influence the distribution such as (age of) children in the household, year or living region. Clear differences occur, for instance, between the eastern and western states in Germany. The density is particularly of interest, as it makes shifts in probability mass or bimodalities due to subgroups easily visible, and as it extends well to discrete, continuous, mixed or bivariate distributions. Such analyses require methods that can model the whole distribution flexibly depending on covariates without parametric assumptions such as normality, and that allow for linear, nonlinear and random effects. We will address this by developing suitable methods for density regression. Depending on the data situation, density regression methods are required and will be developed for density-valued data - e.g. when a histogram is provided for administrative data or is used to summarize massive data - as well as for individual scalar data, when interest lies in modeling the conditional density given covariates. In our unified approach, these two scenarios present two sides of the same coin instead of referring to two different branches of statistics. We will develop methods for continuous densities (e.g. income), discrete densities, i.e. compositional data (e.g. time use over discrete categories), as well as mixed densities (e.g. a woman's fraction of household labor income). Additionally, we will extend this to bivariate densities e.g. when jointly looking at a couple within a household, allowing to overcome also restrictive correlation assumptions.