Additive Functional Regression for Densities as Responses
Additive Functional Regression for Densities as Responses
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DOI：
10.1080/01621459.2019.1604365
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发表时间：
2020-04
影响因子：
3.7
通讯作者：
Kyunghee Han;H. Müller;B. Park
中科院分区：
文献类型：
--
作者：
Kyunghee Han;H. Müller;B. Park
摘要提出并研究了一种新的加性函数回归模型--加性密度回归模型，它适用于响应为随机分布且预测因子为向量的情形。在统计分析中，越来越多地遇到密度或分布样本形式的数据，因此需要灵活的回归模型，将随机密度作为回应。这类模型对多变量连续预测者特别感兴趣，其中不受限制的非参数回归方法受到维度诅咒的影响。可以预期，附加模型将保持一维的收敛速度，同时允许相当程度的灵活性。这促使了在多变量连续预测因子与密度作为响应耦合的情况下的加性回归模型的发展。为了克服分布不能形成向量空间的问题，我们利用一类将密度映射到无限制平方可积函数的变换，然后采用加性函数回归模型来拟合无限制空间中的响应，最后转换回密度空间。我们用光滑回代的扩展版本实现了所提出的加性模型，并建立了该方法的一致性，包括收敛速度。通过对美国婴儿名字分布的应用，说明了所提出的方法。
Abstract We propose and investigate additive density regression, a novel additive functional regression model for situations where the responses are random distributions that can be viewed as random densities and the predictors are vectors. Data in the form of samples of densities or distributions are increasingly encountered in statistical analysis and there is a need for flexible regression models that accommodate random densities as responses. Such models are of special interest for multivariate continuous predictors, where unrestricted nonparametric regression approaches are subject to the curse of dimensionality. Additive models can be expected to maintain one-dimensional rates of convergence while permitting a substantial degree of flexibility. This motivates the development of additive regression models for situations where multivariate continuous predictors are coupled with densities as responses. To overcome the problem that distributions do not form a vector space, we utilize a class of transformations that map densities to unrestricted square integrable functions and then deploy an additive functional regression model to fit the responses in the unrestricted space, finally transforming back to density space. We implement the proposed additive model with an extended version of smooth backfitting and establish the consistency of this approach, including rates of convergence. The proposed method is illustrated with an application to the distributions of baby names in the United States.