We propose a distributional outcome regression (DOR) with scalar and distributional predictors. Distributional observations are represented via quantile functions and the dependence on predictors is modelled via functional regression coefficients. DOR expands existing literature with three key contributions: handling both scalar and distributional predictors, ensuring jointly monotone regression structure without enforcing monotonicity on individual functional regression coefficients, providing a statistical inference for estimated functional coefficients. Bernstein polynomial bases are employed to construct a jointly monotone regression structure without over-restricting individual functional regression coefficients to be monotone. Asymptotic projection-based joint confidence bands and a statistical test of global significance are developed to quantify uncertainty for estimated functional regression coefficients. Simulation studies illustrate a good performance of DOR model in accurately estimating the distributional effects. The method is applied to continuously monitored heart rate and physical activity data of 890 participants of Baltimore Longitudinal Study of Aging. Daily heart rate reserve, quantified via a subject-specific distribution of minute-level heart rate, is modelled additively as a function of age, gender, and BMI with an adjustment for the daily distribution of minute-level physical activity counts. Findings provide novel scientific insights in epidemiology of heart rate reserve.

翻译：我们提出了一种分布式结果回归方法，该方法可同时处理标量预测变量和分布式预测变量。通过分位数函数表示分布式观测值，并利用函数回归系数建模预测变量的依赖关系。分布式结果回归在现有文献基础上有三大创新：能够处理标量预测变量和分布式预测变量，在不强制单个函数回归系数单调的前提下确保回归结构的联合单调性，并为估计的函数系数提供统计推断。采用伯恩斯坦多项式基构建联合单调回归结构，避免过度限制单个函数回归系数需满足单调性的要求。开发了基于投影的渐近联合置信带及全局显著性统计检验方法，用于量化估计函数系数的统计不确定性。模拟研究验证了分布式结果回归模型在准确估计分布效应方面的良好性能。我们将该方法应用于巴尔的摩老龄化纵向研究890名参与者的连续监测心率和身体活动数据。通过个体特异性分钟级心率分布量化的每日心率储备，被建模为年龄、性别、BMI的加性函数，并调整了每日分钟级身体活动计数分布。研究结果为心率储备的流行病学提供了新颖的科学见解。