Flexible density regression methods

这个项目的目标是开发一个灵活的密度半参数回归方法的统一框架，以更好地描述和理解感兴趣的变量之间的关系。虽然(标量)数据的传统面向均值的参数建模的局限性促进了对其他分布特征(分位数)或多分布参数(分布回归)的各种扩展，但最近在作为统计分析对象的概率密度的泛函和成分数据分析中发展了第一种方法。然而，到目前为止，这些具有灵活分布(个体水平方法)和分布统计分析(密度水平方法)的回归分支都是独立开发的。我们的使命是与他们一起进行卓有成效的相互充实，促进方法的发展。为了说明灵活密度回归方法的需求，我们引用了性别经济学中的一个例子：在性别认同规范的问题中，妇女在夫妇劳动收入中的份额的分布是重要的。作为一种概率分布，它在[0，1]上呈现混合分布，正概率质量在0和1处(对于单收入夫妇)和介于两者之间的连续-通常是双峰密度(对于双收入夫妇)。因此，将这种分布与可能影响分布的变量联系起来是很有意义的，例如家庭、年份或居住地区的儿童(年龄)分布。例如，德国东部和西部各州之间出现了明显的差异。密度特别令人感兴趣，因为它使子群引起的概率质量或双峰性的变化很容易看到，并且它很好地扩展到离散、连续、混合或二元分布。这样的分析需要能够灵活地根据协变量对整个分布进行建模的方法，而不需要正态等参数假设，并且考虑到线性、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--以及对于单个标量数据，当感兴趣的是对给定协变量的条件密度进行建模时。在我们的统一方法中，这两种情况呈现了一枚硬币的两面，而不是涉及统计的两个不同分支。我们将开发连续密度(例如收入)、离散密度(即成分数据)(例如离散类别的时间使用)以及混合密度(例如妇女在家庭劳动收入中所占比例)的方法。此外，我们将把它扩展到双变量密度，例如，当联合查看一户内的一对夫妇时，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.