Abstract

This note examines, at the population-level, the approach of obtaining predictors $\tilde{h}(\mathbf{X})$ of a random variable Y, given the joint distribution of $(Y,\mathbf{X})$, by maximizing the mapping $h\mapsto \kappa (Y,h(\mathbf{X}))$ for a given correlation function $\kappa (·,·)$. Commencing with Pearson’s correlation function, the class of such predictors is uncountably infinite. The least-squares predictor $h^{*}$ is an element of this class obtained by equating the expectations of Y and $h(\mathbf{X})$ to be equal and the variances of $h(\mathbf{X})$ and $\mathbf{E}(Y|\mathbf{X})$ to be also equal. On the other hand, replacing the second condition by the equality of the variances of Y and $h(\mathbf{X})$, a natural requirement for some calibration problems, the unique predictor $h^{**}$ that is obtained has the maximum value of Lin’s (1989 Lin, L. (1989), “A Concordance Correlation Coefficient to Evaluate Reproducibility,” Biometrics, 45, 255–268. DOI: 10.2307/2532051.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) concordance correlation coefficient (CCC) with Y among all predictors. Since the CCC measures the degree of agreement, the new predictor $h^{**}$ is called the maximal agreement predictor. These predictors are illustrated for three special distributions: the multivariate normal distribution; the exponential distribution, conditional on covariates; and the Dirichlet distribution. The exponential distribution is relevant in survival analysis or in reliability settings, while the Dirichlet distribution is relevant for compositional data.

摘要

本说明在人口层面检查了获取预测变量的方法$\stackrel{～}{H}(\mathbf{X})$的随机变量Y，给定联合分布$(是,\mathbf{X})$，通过最大化映射$H\mapsto \kappa (是,H(\mathbf{X}))$对于给定的相关函数$\kappa (·,·)$. 从 Pearson 的相关函数开始，这种预测变量的类别是无穷无尽的。最小二乘预测器$H^{*}$是此类的一个元素，通过将Y的期望与$H(\mathbf{X})$相等且方差为$H(\mathbf{X})$和$\mathbf{乙}(是|\mathbf{X})$也是平等的。另一方面，用Y的方差相等代替第二个条件和$H(\mathbf{X})$，一些校准问题的自然要求，唯一的预测器$H^{**}$得到的具有 Lin 的最大值（1989 Lin, L. ( 1989 年)，“评估再现性的一致性相关系数”，生物识别，45, 255 – 268。DOI：10.2307/2532051。[Crossref], [PubMed], [Web of Science ®]  , [Google Scholar] )在所有预测变量中Y一致性相关系数 (CCC)由于 CCC 衡量的是一致程度，因此新的预测器$H^{**}$称为最大一致性预测器。这些预测变量针对三种特殊分布进行了说明：多元正态分布；指数分布，以协变量为条件；和狄利克雷分布。指数分布与生存分析或可靠性设置相关，而 Dirichlet 分布与成分数据相关。